The Welsh actor told the Radio Times that the BBC sci-fi drama "seems like a very good, fun job to be involved in".
"It's one of those roles that has lived through time," Barnard said of potentially playing the 12th Doctor.
"It's an endless tale and what's interesting about Doctor Who is it can go in any direction at any point and every Doctor is taken very differently."
The 26-year-old added that the lead role in Doctor Who is a part "you'd have to consider".
"It's a great piece of storytelling and you can have a lot of fun with that," he continued. "I'm a fan, too, and I do catch it when I can, and I like all the old stuff."
Doctor Who returns to BBC One on Saturday, November 23 for its 50th anniversary special.
> Helen Mirren on Doctor Who: 'Gay, black, female Doctor would be best'
Watch Digital Spy's Geek TV special about Matt Smith's exit and the 12th Doctor below: