The suspect, a man in his 50s from the York area, was arrested by North Yorkshire Police on Monday in connection with Miss Lawrence’s 2009 disappearance and was being held in custody.
Police have urged the public not to name the man on social media as the case is still active.
A police spokesman said that searches at a linked property, in a York cul-de-sac, were completed on Monday night, and that Miss Lawrence, who was 35 when she disappeared, had not been found.
North Yorkshire Police began reviewing the case in 2013. A 60-year-old man was arrested last year in connection with her disappearance and suspected murder but was released without charge, while a 47-year-old man remains on bail on suspicion of perverting the course of justice.
Last week police released previously unseen CCTV of two people walking near Miss Lawrence’s home on the evening she went missing.
Anyone with information should call 101, select option 1, and quote “Claudia Lawrence”.