The move would come as a relief to many Yorkshire businesses, which make up billions of pounds of the region's economy.
Speaking at the daily Downing Street briefing, Oliver Dowden said: “I would love to get the tourism sector up as quickly as we possibly can. We’ve set this very ambitious plan to try and get it up and running by the beginning of July.
“Clearly, we can only do it if it’s safe to do so because I think the worse thing for our tourism sector would be to start, then see the R rate rise out of control, see a second peak that overwhelms the NHS that we then have to slam on the brakes again.
“But, believe me, when we get to the point when we can have British tourism back, perhaps apart from the Prime Minister you won’t get a bigger champion of the great British break than me.”
Tourism is worth £9bn a year to Yorkshire, supporting 224,000 workers – 8.5 per cent of all jobs in the region.
And Welcome to Yorkshire chief executive James Mason said last month he was aware of “dozens” of companies going out of business already, with the number was increasing on a daily basis.
Businessman Malcolm Weaving, who runs The Rendezvous Hotel in Skipton, revealed his hotel had lost over £440,000 as a direct result of the crisis causing 3,500 cancelled bookings.
The move to reopen the industry would put businesses such as B&Bs and venues on the same course as other hospitality businesses that the Government is hoping to open from July 4, if safe to do so, in the third step of their coronavirus recovery plan.
Mr Dowden also set out details of the taskforce being set up to help the arts, sports and digital sectors respond to the crisis.
He told the Downing Street press conference: “Finding creative, crowd-free ways to navigate coronavirus is the biggest challenge for our recreation and leisure sectors right now.”
The taskforce will involve former footballer Alex Scott, former BBC and ITV chairman Lord Grade, English National Ballet artistic director Tamara Rojo and tech entrepreneur Baroness Lane-Fox.
Mr Dowden said: “Normal life, as we have known it, is still clearly a long way off and the path to get there is a narrow one.
“But these things will return, when it’s safe for them to do so, and thanks to the same drive and creativity that makes a great performance or a great piece of art.
“I really think that when they do, and when we have overcome this crisis together, we will appreciate them that much more.”
He also said he hoped some Premier League games would be shown on free-to-air television if the football season resumes.
Guidance will be issued later this week about a return to contact training for elite sports and Mr Dowden has earmarked mid-June as a possible date for the return of the Premier League behind closed doors.
He said that while existing broadcast rights had to be respected, there was the possibility of flexibility because the previously protected Saturday afternoon kick-offs could now be televised as there was no possibility of them keeping spectators away from stadiums.
“I think that creates an opportunity for us to be able to get some sport, some Premier League free-to-air,” he told the Downing Street press conference.
“Those discussions are ongoing.”
Mr Dowden added: “I hope we can sort this out and I also hope then we can get some more money going into the sport of football. I think we can find ourselves in a win-win situation.”
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