The newspaper quotes an internal memo at Finsbury that says the interview related to a "share-dealing investigation". The driver "has not been charged with any wrongdoing, and there is no suggestion that he dealt or profited personally", it reportedly stated.
Finsbury said in the memo it was "cooperating fully with the FCA's investigation".
City AM reports that the interview is understood to have taken place around eight months ago in relation to a deal involving a Finsbury client last year. Employees at banking and broking groups are also said to be included in the probe.
Finsbury and the FCA both declined to comment.
Rudd founded Finsbury in 1994 and was named the second most powerful UK agency figure in corporate and financial PR by PRWeek earlier this year.
Rudd, who campaigned against Brexit as is part of the pro-EU group Open Britain, is a very well known figure among national newspaper journalists.
Last Friday senior Daily Mail journalist Andrew Pierce published a 1,500-word piece on Rudd, focusing on his work for Open Britain and describing him as an "oleaginous charmer with just a hint of the spiv". In an apparently ironic gesture, Rudd retweeted the article but has since undone the retweet.
WPP-owned agency Finsbury, whose clients include Centrica, RBS and Starbucks, generated global revenue of an estimated $192m in 2017, according to PRWeek’s Agency Business Report.