Miss Lawrence was 35 when she disappeared from her home in York in 2009.
North Yorkshire Police said forensic examinations had been conducted at a house in York, following the man’s arrest.
Forensic officers have boarded up the property as part of their investigations.
North Yorkshire Police said Miss Lawrence, who went missing in March 2009, has not been found.
A force spokesman said: “Police cannot rule out the possibility of further arrests in the future as the review continues.”
The spokesman added: “Forensic examinations and searches are being conducted which are expected to be ongoing over a number of days.
“A car has also been seized as part of the inquiries.
“Claudia’s family have been notified and are being supported by trained officers.”
Police launched a review of the investigation last year after a new major crime unit was established. Detectives have always believed the chef was murdered.
Miss Lawrence, who worked at York University, was reported missing by her father Peter on 20 March, 2009.
She was last seen at just after 3pm on 18 March, walking towards her home, and that night she spoke to both her parents on the phone.
It is thought something happened to her after she left for work early on 19 March.
Last year, a new investigation team conducted a detailed re-examination of Miss Lawrence’s home in the Heworth area of York.
In March, senior officers announced a series of new lines of inquiry on the fifth anniversary of the day she failed to arrive at work. The announcement coincided with a fresh appeal on BBC’s Crimewatch.
Police revealed several new lines of inquiry generated by the review of the case and a fresh examination of her home.
Fingerprints had been uncovered using techniques not available in 2009, they said. They added that new forensic examinations of her house uncovered the fingerprints of people who had not yet come forward.
They also found the DNA profile of an unknown man on a cigarette butt in the chef’s Vauxhall Corsa.
A team led by Detective Superintendent Dai Malyn spent two months re-examining Miss Lawrence’s small terraced home.
Police have said before that Miss Lawrence’s mobile phone and rucksack have never been found.
Last night, the missing chef’s mother, Joan, said she was still “taking in” the news of the arrest after five years of campaigning to try to find her daughter.
The 70-year-old, who is divorced from Claudia’s father Peter, campaigned for a new team to reopen the case after she clashed with the man who led the initial investigation, Detective Inspector Ray Galloway.
He insisted Claudia’s disappearance was linked to her “complex” private life – but Mrs Lawrence refused to accept that.
She said: “I just haven’t had time to take it all in. I’m still trying to get my head around this. There’s been no chance to think it all through. I don’t know what this might mean but it’s not the first time there’s been a big development which has led to nothing. I’m not getting my hopes up just yet.”
A spokesman for Claudia’s father Peter, said that he was “very encouraged” by the news of the arrest.