Mr Capaldi, who is set to make his proper debut as the new Dr Who on Christmas Day, has been put forward for the role on petition website change.org.
The rector at Glasgow is supposed to represent the views of the student body to university management and is elected by students every three years.
To nominate a candidate, students must gain the person’s consent as well as 10 supporting signatures from students at the university.
Fiona Duncan, a fourth-year politics student at Glasgow who organised the petition, said she had yet to contact Mr Capaldi but was hoping interest in the petition would encourage him to stand.
She said: “I’m trying to…show that there is a demand from the student body for him to stand before contacting him. It would be great to have someone from the arts representing Glasgow students.”
If he did choose to run for election, Mr Capaldi would join a long list of actors who have held the position.
Richard Wilson, better known as Victor Meldrew from TV comedy One Foot in the Grave, was rector from 1996 to 1999 and Ross Kemp, who played Grant Mitchell in EastEnders, was in the role from 1999 to 2000 before resigning.
Mr Capaldi, who was born in Glasgow and made his name as foul mouthed spin-doctor Malcolm Tucker in the Thick of It, was described by Ms Duncan as someone “who represents our interests now and what can be achieved in our future”.
Explaining why he backed the campaign, one signatory, Jordan McGlynn from Paisley, wrote: “Because people with menacing stares are more fitting representatives of the populous than those with tame smiles.”
Former leader of the Liberal Democrats Charles Kennedy is the current rector at Glasgow having been elected for a second term in 2011.
The deadline for submitting nominations is 21 January, and the election will be held on either 17 or 18 February.