Local newspapers have named him as Michael Snelling.
The chef was reported missing in March 2009, after failing to turn up to work at the University of York.
A police spokesperson has said: "The investigation team would like to stress that Claudia has not been found. Police cannot rule out the possibility of further arrests in the future as the review continues."
The home of the man who is being held in custody has been sealed off by police and forensic teams have been seen going in and out of the property throughout the day.
According to reports from the scene, activity is currently focused on the back-garden of the property and on a paved area behind the garage.
Evidence bags and a suitcase have been removed from the house.
A second property, in North Shields, Tyneside, is also being searched and is believed to be linked to the investigation.
In March, Claudia’s case formed part of a Crimewatch special, after detectives involved in the case revealed they had ‘new lines of enquiry’ which they hoped would lead them to answers.
Detectives reported how her mobile phone was deliberately turned off on the day she went missing and also explained that unidentified fingerprints had been found in her home, on Heworth Road, along with a male’s DNA on a cigarette butt, found in her car.
Claudia, who was 35 when she went missing, was last seen at around 3.05pm on 18 March, walking towards her house.
Claudia's family have campaigned for five years to find out what happened to her, with her father, Peter, telling Sky News: "It's actually, I suppose, a bit like a cancer and it just gets worse and worse because there is no resolution until we know what happened to Claudia."
More to follow.