The force yesterday said it had asked the BBC’s report into Clarkson’s unprovoked attack on producer Oisin Tymon at Simonstone Hall near Hawes.
But it said today in a statement that although no-one present at the time of the incident had reported an offence, it had a “duty to investigate where we believe an offence might have been committed”.
The statement said: “Over the past few days, we have been contacted by concerned members of the public, who have seen media coverage about Jeremy Clarkson’s actions at Simonstone Hall, and who want to know what action the police are taking.
“No-one who was present at the hotel during the incident came forward to report an offence to the police. Nonetheless, we have a duty to investigate where we believe an offence might have been committed, and that is what we have been doing with this case.
“We have already spoken to some people who were in the hotel at the time, but there are still others we need to speak to, who could help us to determine exactly what happened.
“Also, as is usual in these circumstances, we have made contact with Mr Tymon through his lawyers, to ask him to speak to us so we can ascertain how he wishes to pursue this matter.
“When these routine enquiries are completed, we will be able to assess whether or not further action needs to be taken from a police perspective.
“Meanwhile, we’d like to reassure the public that the matter is in hand, and we dealing with it responsibly and proportionately.”
Clarkson, who left his west London home on a bicycle shortly before 1pm, faces an uncertain future as reports suggest the former Top Gear host’s co-stars may leave the show alongside him.
He did not stop to answer reporters’ questions, but when asked what he would like to say to fans who may be upset about him leaving the show, he said: “Everybody’s upset.”
BBC director-general Tony Hall has made it clear he wants the show to continue but its future is in doubt after co-host James May said he was considering his position and Richard Hammond declared the “end to an era”.
The BBC confirmed Clarkson’s contract would not be renewed after its investigation found the presenter’s attack on producer Oisin Tymon left him needing hospital treatment.
Mr Tymon, who did not report the incident and thought he had lost his job, has also been subject to abuse from Top Gear fans on social media.
Clarkson’s immediate concern will be possible criminal action after North Yorkshire Police asked to see the BBC report into the 30-second physical attack at a hotel in the county.
A force spokeswoman said: “The information will be assessed appropriately and action will be taken by North Yorkshire Police where necessary. It would not be appropriate for North Yorkshire Police to comment further at this time.”
As for the 54-year-old’s broadcasting future, two major broadcasters, Sky and Channel 4, are understood to have ruled out working with him, while ITV would not comment on “a BBC issue”.
But the BBC’s creative director Alan Yentob offered a potential olive branch by refusing to rule out a return to the corporation for Clarkson in the future.
He told BBC2’s Newsnight: “Of course I wouldn’t rule that out, but that’s not for now.”
All three presenters’ contracts run out at the end of this month.
May described Clarkson’s departure as a “tragedy” but said: “I’m sure Top Gear will continue in some way. It existed before us and it has been reformatted several times.”
Asked if he will stay at Top Gear, he added: “I don’t want to talk about that too much but I think we are very much the three of us as a package. It works for very complicated reasons that a lot of people don’t fully understand. So that will require a lot of careful thought.”
Hammond said he was “gutted at such a sad end to an era” after Clarkson’s sacking. He added: “We’re all three of us idiots in our different ways but it’s been an incredible ride together.”
The BBC declined to comment on the future of the other two hosts. Mr Yentob acknowledged it was “perfectly natural” for Hammond and May to have reservations about continuing Top Gear but said he would be disappointed if they quit.
He said: “It’s perfectly natural for them. They are a team they have worked together for a very long time, they are all very attached. The question of what happens next for Top Gear is a conversation which must go on between the controller of BBC2 Kim Shillinglaw, between Andy (Wilman, producer of Top Gear) and the team.”
Chris Evans leads the betting as favourite to take over at the show even though he has ruled himself out on more than one occasion.
In his last column for the show’s magazine, Clarkson hit out at Top Gear’s detractors and challenged them to make their own rival show.
He wrote the piece before the attack on Mr Tymon but it was only published today.
In something of a coincidence, the star, who may be mulling over ideas for a new motoring show to take to another broadcaster, told unhappy viewers to “get off your arse and help yourself”.
He told the programme’s critics, who he called “hopeless worms”, that instead of moaning that Top Gear should “go back to the days when it reviewed cars” they should try making their own show.
In the magazine, on sale today, the broadcaster said there was plenty of room in the market for another car show.
“You have Top Gear and Fifth Gear and Chris Harris and, er, that’s it. So, in order to shine, all you have to be is better than Captain Slow, a midget, a pensioner, an orangutan and a monkey. Which isn’t hard,” he said.
This morning staff from a car and van rental company pulled up outside Clarkson’s home with a banner over their vehicle which read: “Are you a good driver? Are you out of work? Need new delivery drivers - apply now.”
Bookmaker Paddy Power has set up a blackboard at the front of Clarkson’s home outlining the odds for who will be the next Top Gear host and what is next for Clarkson.
Chris Evans is listed as the evens favourite followed by Jodie Kidd at 10/3, Guy Martin at 5/1, Stephen Fry at 7/1, Piers Morgan at 10/1, Ant and Dec at 20/1, Simon Cowell at 50/1, Boris Johnson also at 50/1, and Lewis Hamilton and Zayn Malik both at 100/1.
The odds that Clarkson will join Sky are 4/1, join ITV 5/1, open a steak joint 6/1, be arrested 7/1, join Ukip 16/1, join the Conservative Party 40/1, release a fitness DVD 50/1, replace Zayn in One Direction 100/1, be the Mayweather undercard 250/1 and become Argentine ambassador 500/1.
Prime Minister David Cameron was asked about the presenter’s departure from the BBC during a visit to the London Taxi Company’s factory in Coventry.
Asked if the BBC had made the right decision concerning “one of Britain’s great automotive exports” in the form of Clarkson, Mr Cameron responded: “It’s entirely a matter for the BBC.
“It’s never right not to treat your staff and people properly. So it’s their decision and I don’t think I should say anything more about it.”
Oisin Tymon has come under huge abuse from trolls on Twitter, including receiving death threats.
The Twitter abuse towards Mr Tymon is violent and threatening, with many blaming him for Mr Clarkson’s departure from Top Gear.
Chris Brennan (@chrisbrennan10) wrote: “So Oisin Tymon got a slap for being lairy then drove himself to A&E then squealed like a b**** That’s a coward right there.”
Micha Galgentanz (@Galgentanz) tweeted: “Oisin Tymon - hope you’ll get screwed for the rest of your useless life.”
Gonzalo Robles (@grobles17) wrote: “Clarkson should have killed you Oisin Tymon, filthy b******.”
Paul Richer (@PaulmRicher) wrote: “Oisin Tymon had to go to A&E !!! let’s hope he visits the morgue VERY soon , a***wipe !!!” and “Tony Hall BBC director , I wonder if Oisin’s and your head can stop a bullet !!! just wondering.”
Pasan Weerasinghe (@PortelloFanatic) wrote: “To meet Oisin Tymon in person and feed him the very food he denied Clarkson. Then, beat him to a pulp with a rolling pin. LifeGoals.”
Maurice Nixon (@ace197777) wrote: “Hope u never work again (Oisin Tymon) n end on the streets u waste of space.”
Darren Laws (@01lawsda) tweeted: “You are a f****** c***, may you have a very slow and painful death OisinTymon.”
Many Twitter users wrote of their surprise and anger at those trolling Mr Tymon.
Ollie Connors (@olliexcore) wrote: “I wonder how the people out for Oisin Tymon’s blood would feel if their boss punched them for no good reason?”
Louisa Booth (@louisabooth) tweeted: “Man assaults another man and victim receives abuse because people can’t watch a TV show about cars. Bravo society. Bravo OisinTymon”.
Jason Manford, the comedian, wrote: “It’s surprising how sensitive fans of someone so controversial can be. Clarkson.”