North Yorkshire Police arrested the suspect yesterday in connection with Miss Lawrence’s disappearance in 2009.
A spokesman said searches at a linked property were completed last night. Yesterday, police could be seen searching a semi-detached house in a quiet cul-de-sac in York.
The spokesman said yesterday that Miss Lawrence, who was 35 when she went missing, had not been found.
Miss Lawrence’s family have been told about the arrest and are being supported by trained officers.
Detective Superintendent Dai Malyn, of North Yorkshire Police, urged people not to identify the bailed man, who is from the York area.
He said: “To ensure the investigation and legal process are not compromised or potentially damaged in any way during this critical phase in seeking the truth about Claudia’s disappearance, North Yorkshire Police strongly advises the media and members of the public against identifying the man.
“This includes naming or publishing images of the man on traditional media platforms or social networking sites.
“I urge everyone to show restraint and patience while we carry out these very important inquiries.”
Miss Lawrence was reported missing by her father 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 60-year-old man was arrested last year in connection with her disappearance and suspected murder but was later released without charge, while a 47-year-old man remains on bail on suspicion of perverting the course of justice.
Mr Malyn said he was “actively pursuing new leads” and that his team had made “significant progress” since the force began reviewing the case in 2013.
The detective said some people locally knew Miss Lawrence but have kept their relationship secret and some deliberately lied about a number of issues concerning their association with the chef.
Last week, on the sixth anniversary of her disappearance, police released previously unseen CCTV footage of two people seen walking near her home on the evening she went missing.
Miss Lawrence’s father, Peter, said yesterday: “It is encouraging to know that, following all the media activity over the last three weeks, from Claudia’s 41st birthday to the sixth anniversary of her being missing, North Yorkshire Police continue to be active in seeking answers as to what has happened to Claudia.
“It is to be hoped that the matter can be resolved as soon as possible and I encourage people to continue to come forward with information to the police.”
Last week, previously unseen CCTV footage of two people walking near missing chef Miss Lawrence’s home on the evening of her disappearance was released by detectives.
North Yorkshire Police appealed to local residents for help to identify the people seen in the footage, which was screened on a ‘digivan’ near Miss Lawrence’s house in York on the sixth anniversary of her disappearance.
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. Please quote “Claudia Lawrence” when passing on details.