The suspects, who have not been named by North Yorkshire Police, are in their 50s and from the York area. They were arrested yesterday morning and taken into custody for questioning.
In a statement today, the force said they had been released “on police bail in order to allow detectives time to follow up lines of enquiry”.
Detectives said yesterday that searches were being conducted and “are expected to be ongoing over a number of days” but have declined to release further details.
Yesterday morning, two houses nearby to Claudia’s were being searched by police, who cordoned off half of one street with crime scene tape.
The other was being flanked by two police officers while officers searched inside. Two police vans were parked outside each each house.
A resident living near one of the houses said police arrived at around 7am this morning.
Tommy Ward, 81, said: “It started off with one police car then a few more came. It was just around about 7am when everything started up.
“I was outside my house watching the world go by at the time. I haven’t seen anyone arrested but a few weeks ago some sniffer dogs were there in the same house. We’ve had dogs a few times now.
“I don’t know the person who lives there.”
The arrests brings the total number of arrests to six since North Yorkshire Police launched a review of its original Claudia Lawrence investigation.
A 60-year-old man was arrested last year in connection with her disappearance and suspected murder but was later released without charge.
Another local man in his 50s was arrested last month on suspicion of murder, before being released on bail while enquiries continue.
A man arrested last year on suspicion of perverting the course of justice was released from his bail earlier this month.
The force said the decision to release the 47-year-old from his bail conditions was made after consulting with prosecutors and “following information the man provided to us which has progressed the investigation and clarified certain issues”.
It is understood the three men arrested today have not previously been arrested as part of the Claudia investigation.
Miss Lawrence’s father, Peter Lawrence, has led a tireless campaign to find out what happened to his daughter.
His friend and spokesman, Martin Dales, said all new news is difficult for the family.
Speaking outside Miss Lawrence’s house, Mr Dales said: “It’s happened so many times over the years but it never gets any less to deal with. It’s still an emotional shock.”
He thanked members of the public for continuing to supply the police with information.
“The key message today is that police have made it quite clear that there are people lying to them.
“And it’s those people who’ve just got to realise what it’s like after all this length of time - family and friends waiting to find out what’s happened and you’ve got that information and you’re too selfish to give it, or you’ve got something to hide, or a skin to preserve.”
Miss Lawrence was reported missing by her father in March 2009 after concerns were raised when she failed to turn up for her 6am shift at work.
North Yorkshire Police began reviewing the case in 2013 and have since carried out a number of searches, including a detailed re-examination of Miss Lawrence’s home, in the Heworth area of the city, and a fingertip search of an alleyway that leads to the rear of the house.
A spokesman for North Yorkshire Police said today: “The investigation team would like to stress that Claudia has not been found. Claudia’s family have been notified about the arrests and are being supported by trained officers.”
Detective Superintendent Dai Malyn, head of North Yorkshire Police’s Major Crime Unit and the senior investigating officer in the Claudia Lawrence inquiry, said: “As with the previous arrest last month, I strongly advise the media and members of the public against identifying the men who have been arrested.
“This is to ensure the investigation and legal process are not compromised or potentially damaged in any way.
“This includes naming or publishing images of the men, their home addresses, places of work, their vehicles and properties being searched on traditional media platforms or social networking sites.
“I urge everyone to again show restraint and patience while we carry out these very important enquiries. Your co-operation and support is greatly appreciated by the investigation team.”
Anyone with information that could assist the investigation should contact North Yorkshire Police on 101, select option 1, and pass details to the Force Control Room.