The Death in Paradise star, 44, was an early favourite to replace Peter Capaldi, 59, who quit as the Time Lord in January.
But after missing out on the top role to Jodie Whittaker, Kris is the main contender to succeed companion Pearl Mackie, 30, who is leaving after just one series.
If successful, he may join up with Matt Lucas, 43, who plays the other assistant Nardole.
A source said: “Kris is a big fan of the show and the BBC are a big fan of his.”
Kris, who shot to fame as Nick Harper in My Family, quit Beeb show Death in Paradise this year after four series.
A BBC spokeswoman said there had been no decisions yet on companion casting.
Hello. I'm Lewis Moon. This is the blog with all the doctor who news as well as MLMA updates.
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