Jenson Button with his wife Jessica Michibata
Button, who married model Jessica Michibata last December, is friendly with Evans and has twice appeared on Top Gear before Jeremy Clarkson, Richard Hammond and James May left.
Neither Button’s spokesman nor the BBC are said to have denied that talks are taking place, but declined to comment on speculation.
Evans, the Radio 2 DJ, is replacing Jeremy Clarkson as the lead presenter.
Button in the reasonably priced car
The known car enthusiast has signed a three-year deal to head up the BBC Two show, calling it his “favourite programme of all time”.
Others touted to join him include Jodie Kidd, Guy Martin, Philip Glenister and Dermot O’Leary.
Evans has vowed not be offensive for the sake of it, suggesting that Clarkson had to resort to shock tactics due to a “lack of imagination”.
He is expected to be paid at least £1 million a year for the role and will also be executive producer.