There is speculation that the Scottish actor will depart in 2017 at the same time as current showrunner Steven Moffat, to allow incoming chief Chris Chibnall to choose his own Doctor.
The Mirror quoted an unnamed BBC source, who said: “Bosses are already discussing a fresh start when Steven leaves.
“They’ve been happy with what Peter has brought to the role but some think it might make sense to give Chris his own choice of actor to play the Doctor.
“They hope he will be able to find someone who can have the same impact as Matt Smith did when Steven Moffat kicked off his run as executive producer.”
The departure of the Doctor at the same time as the showrunner would mirror what happened with Russell T Davies who departed with 10th Doctor David Tennant.
A BBC spokeswoman declined to discuss Capaldi’s contract but confirmed he was signed up for the duration of series 10.
Broadchurch creator Chibnall was announced as head writer and executive producer on Saturday following Moffat’s decision to step down after six years at the helm.
A Doctor Who Christmas special will air later this year, while series 10, Moffat’s last, is scheduled to be broadcast in 2017, with a new companion on board to replace Jenna Coleman.
Moffat said of his departure: “Feels odd to be talking about leaving when I’m just starting work on the scripts for season 10, but the fact is my timey-wimey is running out.
“While Chris is doing his last run of Broadchurch, I’ll be finishing up on the best job in the universe and keeping the Tardis warm for him.
“It took a lot of gin and tonic to talk him into this, but I am beyond delighted that one of the true stars of British television drama will be taking the Time Lord even further into the future.”
He joked: “At the start of season 11, Chris Chibnall will become the new showrunner of Doctor Who. And I will be thrown in a skip.
Chibnal paid tribute to Moffat, saying: “He’s been a dazzling and daring showrunner, and hearing his plans and stories for 2017, it’s clear he’ll be going out with a bang. Just to make my life difficult.”
The report has already got fans speculating about who could be named as the next Doctor.
Comedian, director and star of sitcom The IT Crowd Richard Ayoade is one name that has been mooted.
Harry Potter favourite Rupert Grint is another star some fans have thrown into the hat. The 27-year-old former child star would actually be older than fan-favourite Matt Smith was when he took on the role.
Trainspotting star Robert Carlyle could continue a long Scottish association with Dr Who, following in the footsteps of Capaldi himself, as well as David Tennant and Sylvester McCoy.
Some fans believe the time is right for a female Doctor, with Actress Hayley Atwell having reportedly thrown her hat in the ring on Twitter last year.
Who would you like to see take up the mantle?