According to the Daily Mirror, the actor, comedian and quiz show host, 57, will accompany Jodie Whittaker, 35, who was recently unveiled as the first female Time-Lord.
"Bradley is super excited to be joining the cast of Doctor Who in such a key role," a source was quoted as saying.
"It means that his schedule over the coming months will be jam-packed".
Walsh, who starred in Coronation Street and recently enjoyed success with a singing career, will continue to present ITV quiz show The Chase, according to the report.
Some of the drama's fans love the prospect of the funnyman in the role.
Actor Matt Lucas, himself a companion on the show to Peter Capaldi's Doctor, wrote on Twitter: "No idea if the story about Bradley Walsh being in Doctor Who is true, but if it is then I love it. #TeamBradley".
@GarHop78 wrote: "Bradley Walsh as a companion in Doctor Who is so ridiculously out of the box it might just work."
@PaulGold92 added: "I've never wanted anything to be truer".
Others were more unsure.
@Haych_Pea said: "The BBC really are struggling for funds if Bradley Walsh in set to be the new Doctor Who companion."
And @dannycharleston wrote: "Can everyone please stop acting like Bradley Walsh as the Doctor Who companion is a good idea?"
@askyerfather wrote: "This is wrong isn't it."
Previous companions have been played by Billie Piper, Freema Agyeman, Jenna Coleman and most recently, Pearl Mackie.
Broadchurch star Whittaker will take over from Capaldi as the 13th reincarnation of the Time Lord.
Whittaker told fans not to "be scared by my gender" and said: "It feels completely overwhelming, as a feminist, as a woman, as an actor, as a human, as someone who wants to continually push themselves and challenge themselves, and not be boxed in by what you're told you can and can't be. it feels incredible."
A BBC spokesman refused to comment on the Walsh speculation.
Walsh's representatives have also been contacted over the report.