The Green Wing star is said to be among BBC producers' list of actors being considered for the next Time Lord, following Matt Smith's departure at Christmas.
Rhind-Tutt was reportedly one of the last few contenders to be The Doctor after David Tennant left in 2010, according to The Sun.
Torchwood actor Burn Gorman is also apparently one of the final contenders for the role.
Threesome's Emun Elliott and Richard Coyle - who worked with Doctor Who writer Steven Moffat on Coupling - are also thought to be on the wishlist.
Rory Kinnear is now said to be the favourite to take on the role, according to the tabloid.
Homeland actor David Harewood has said that it would be "hard" to turn down the part.
Dominic Cooper, Ben Daniels and Chris Addison have also been linked with the role in recent weeks.