Sir Robert Goodwill sent the email to around 20 people who repeatedly asked him to publicly condemn Mr Johnson for breaching coronavirus laws with his 56th birthday celebration in 2020.
The MP for Scarborough and Whitby told them there is “a whole mine of talent within the current cabinet who could potentially take over Number 10” and it “would be helpful if you could let me know which of the alternatives you believe would make the best Prime Minister".
In the email, he said Chancellor Rishi Sunak has "shown tremendous leadership and judgement during the pandemic", Deputy Prime Minister Dominic Raab has "shown he can step into the breach when the Prime Minister is away" and Health Secretary Sajid Javid has got "tremendous skills and an amazing tack record".
But Sir Robert told The Yorkshire Post he sent the email to “smoke out” those who only want the Prime Ministers to be removed because they support other political parties.
“This wasn't a survey of lots of people to ask who I should support,” said Mr Goodwill.
“The majority of people who came back said they wanted Keir Starmer, so their motivation for calling me to support moves to censure Boris were fairly clear.”
The Tory MP also said he does not want a leadership election, which could take several months, and he will not submit a letter of no confidence.
Sir Robert also said he accepts the Prime Minister’s apology and he thinks the country should “draw a line under it and move on”.
“With the current situation with energy supplies, the cost of living crisis, the refugee crisis and the multitude of other issues we’re facing, having a leadership election would be an indulgence which would not serve the country well,” he said.
“We're behind the Prime Minister on his policy on sending economic migrants to Rwanda to have their claims looked at, we're behind the Prime Minister and the Chancellor on their economic policy and we very much think the Prime Minister has done the right thing in terms of Covid.”
He added: “I think that, given there are no major policy issues and there are no major delivery issues in terms of the Covid pandemic and supporting Ukraine, I really don't think it's a good idea for us to have the distraction of a leadership election.”
Labour has tabled a motion calling for the Prime Minister to be referred to the Commons Privileges Committee over allegations he was untruthful when he assured the Commons that coronavirus regulations were followed in Downing Street during the pandemic.
Mr Johnson has since been fined by Metropolitan Police officers for his part in a Cabinet Room birthday bash in June 2020, with more fixed-penalty notices a possibility as the inquiry into potential Covid-19 regulation breaches at the heart of Government continues.
Sir Keir Starmer’s party is calling on Conservative MPs to back its call for a Privileges Committee probe during a vote on Thursday, threatening to make those who support Mr Johnson the subject of campaign advertising.
Without Tory votes, the motion, which was published on Wednesday, is destined to fail, despite being backed by figures from six other opposition parties.