The MP - who has been a staunch supporter of Boris Johnson on many controversial issues in the past - says both Boris Johnson and Rishi Sunak should "do the right thing" and step down because they broke the rules their cabinet had put in place.
Speaking during his weekly Facebook Q and A this evening, Mr Whittaker said: "I not only think that the Prime Minister should resign but I also think that Rishi Sunak should resign as well.
"Through this whole process it hasn't been particularly clear that the Prime Minister broke any rules until of course he's been issued with a fixed penalty notice this week.
"My expectation is that he and the chancellor should do the right thing and resign.
"The reality is that they're not going to resign. We've seen that from the press and they've both issued apologies so I suspect we'll end up where we are and moving on."
But Mr Whittaker said he won't be submitting a letter to the 1922 Committee calling for a vote of no confidence in the Prime Minister.
"All that does is, after the trigger point of a number of letters, there is a vote of no confidence in the Prime Minister and my expectation is that he would win that process and with the amount of stuff currently going on, with Ukraine and the crisis around energy and the cost of living crisis for people, I think that's just taking away from what needs to be done on a day-to-day basis and that is to get on and manage the country."
Mr Whittaker is one of the first Conservative MPs to publicly call for the pair to resign.
The Cabinet have backed Mr Johnson and Mr Sunak, and many other Conservative MPs have said they think the pair should stay in office.