The 40-year-old, best known for her role as FBI Special Agent Dana Scully in the sci-fi drama, is expected to be cast opposite the new Doctor Matt Smith.
Doctor Who's makers want the London-based actress to appear in a one-off special episode as a deadly enemy of the Time Lord.
The American star would play the part of the renegade Time Lady The Rani, a glamorous but evil scientific genius previously played in the series by Kate O'Mara in the 80s.
An insider told the Daily Express newspaper: "Gillian obviously has a massive sci-fi following and it's felt that it would be a major coup to have her appear in Doctor Who.
"The team behind the show are keen for the next Doctor to have lots of new enemies and Gillian would be a glamorous and impressive addition to the list.
"The Rani would be a perfect role for her as the character used to be regarded as one of the Doctor's most deadly opponents."
Anderson is currently preparing to star alongside former Doctor Who Christopher Eccleston next month in a production of A Doll's House at the Donmar Warehouse in London's West End.
I ignore a lot of these sillly stories, but this one could happen...Gillian Anderson, best known for X-Files Special Agent Dana Scully, is expected to be cast opposite the new Doctor Matt Smith.
It is claimed the Doctor Who makers want the London-based actress to appear in a one-off special episode as a deadly enemy of the Time Lord.
The American star would play the renegade Time Lady The Rani, a glamorous but evil scientific genius.
OK, it probably still won't happen, but I think its a possible, after all she lives in the UK and having a US actress in the show makes it do well in the USA.