Farming Minister George Eustice, who was Mr Cameron’s former press secretary, believes Mr Cameron is vital to “oil relations” with European leaders in the two years it would take to formally leave the EU.
Boris Johnson will get a significant Cabinet position and the exit negotiations would be led by Mr Cameron, Michael Gove and senior Tory Oliver Letwin, said Mr Eustice.
Forecasting a post-June 23 scenario in which Britain waves goodbye to the EU, pro-Brexit Mr Eustice also said the handful of backbenchers deeply unhappy with Mr Cameron will naturally speak out but will not trigger calls for a leadership contest.
He said: “Paradoxically David Cameron’s position is stronger if the vote is to leave than if it’s to remain.
“Firstly there’s a great deal of sense in having the continuity of David Cameron who has got good relations with other EU leaders. There is a very strong argument for keeping him in place because we will need to oil our relations with our other European counterparts.
“David Cameron has said he is going in two years anyway so to leave him in place to help nurse relations with our European partners makes a great deal of sense.
“Although leaving the EU is not the legacy David Cameron wants, I think it’s not a bad legacy to have.
“He might not realise it yet, but if we do vote to leave and David Cameron stays as a leader during that transition period he would go down in history as the Prime Minister who delivered from this referendum an outcome that history would judge is right for our country.”
Rumours have circulated in Westminster this week that if Britain votes to remain in the European Union, David Cameron will be in danger of a vote of no confidence from angry Conservative backbenchers.
An anonymous Conservative source speaking to the Daily Mail said if the Remain camp win by a small margin then the only way to appease backbenchers is to name the date when Mr Cameron is going to go.
“Would there be a handful of malcontents on the Tory backbenches who have lost it with David Cameron and would like to see him go come what may? Yes. There’s a handful of such people but the party at large won’t be like that,” said Mr Eustice.
“The last thing we want having voted leave is to have the uncertainty of a leadership contest of who should be Prime Minister. That makes no sense.
“You would probably have a duo of Oliver Letwin, who David Cameron trusts deeply, and Michael Gove leading the negotiations. Oliver Letwin because he is a master of detail on every single corner of Government and while he is supporting the vote to remain, Oliver Letwin has always been quite sceptical about the European Union and conscious of its shortcomings and Michael Gove having credibility as he has led the Leave campaign.
“Boris will have a big place in Government at that point. I’m sure there will be quite a big reshuffle.”
Mr Eustice was elected to the Cornish seat of Cambourne and Redruth in 2010 and appointed Farming Minister in 2013.
He had initially been asked by Mr Cameron’s team to help on the Remain campaign, but he said he lost faith in Mr Cameron’s renegotiations in October 2015.
“It was clear to me that we weren’t going to do a renegotiation at all. What we were actually going to do was a choreographed PR exercise to suggest the renegotiation had happened.”
An attempt at to trigger a leadership election immediately after Brexit would not serve any potential candidate well, he said.
“If you were a leadership contender who had been on the Remain side you would want time for people to forget the referendum. If you have been on the Leave side you wouldn’t want to ruin your position by being in a sort of ugly haste to progress things so I don’t think anybody will be pushing for an early leadership contest,” he said.