The 35-year-old Captain America star expressed his desire for the coveted role, saying: "It's an incredible show", adding that he would "absolutely" contemplate the part.
In an interview on This Morning Cooper said that he felt honoured to even be thought of as a possible Doctor: "It would be very scary...but of course I would consider it."
The current Doctor, Matt Smith, is leaving the show at Christmas, but Cooper will have fierce competition should he hope to succeed him, with names such as Bond star Ben Whishaw, and Homeland actor David Harewood rumoured as up for it.
In the meantime the Mamma Mia! star has revealed that he will be returning to appear in the sequel of Captain America. "They are quite a secretive bunch and what you do is just turn up and film it and then you watch it later and find out how it fits in," he said.
On 3 June, 2013, we wrote...
Matt Smith has announced that he will quit as Doctor Who at the end of 2013, and the hunt for his replacement is already underway.
Among the names being suggested is Dame Helen Mirren, who would be the first ever female Doctor Who if she got the part. Speaking about the role she said: "I would like to play the new female Doctor Who. I don't just want to be his sidekick."
Other women being linked with the role include Olivia Colman and Fiona O'Shaughnessy.
John Hurt has also been linked with the role, after being introduced as 'The Doctor' at the end of the last series - and he is due to return to the show for its 50th anniversary episode in November.
It is also thought that the next incarnation of The Doctor could be Idris Elba, who would be the first black actor to play the iconic TV role.
William Hill is currently listing Rupert Grint and Russell Tovey as favourites to score the role, at odds of 10/1.
Show executives are keeping mum on the subject. "Somewhere out there... is someone who's about to become the Doctor," said executive producer Steven Moffat.