Leading Leave campaigners have claimed it would be "disastrous" if a vote to quit the EU sparked a Tory leadership contest.
Brexit-supporting Cabinet ministers insisted David Cameron must remain in No 10 to lead the negotiations pulling Britain out of the 28-member bloc.
It follows a weekend of speculation about the premier's future sparked by former chancellor Ken Clarke's claim that Mr Cameron "wouldn't last 30 seconds" if he loses the June 23 referendum.
Commons Leader Chris Grayling said replacing Mr Cameron would harm the Leave cause.
"I actively want David Cameron to stay, not only because he is a very good prime minister but because he has got the relationships we need around Europe to build a negotiating process," he told BBC One's The Andrew Marr Show.
"It would be disastrous in my view for the Leave cause if we vote to leave and then we get distracted by a leadership contest."
Northern Ireland Secretary Theresa Villiers said Mr Cameron is the "right man to take us out of the European Union" and negotiate a new deal.
She told Murnaghan on Sky News: "I think the Prime Minister's position should be absolutely secure. It's vitally important that he stays regardless of the vote in the referendum because he's a great prime minister."
Senior Tory David Davis told Marr that the Prime Minister would have to appoint someone that the party and the public had faith in to lead the negotiations.
"If he did that, I don't see any reason why he shouldn't go on," he added.
Brexit campaigners continued to dismiss interventions from international leaders and organisations about Britain's future in the EU.
Ahead of the upcoming visit by Barack Obama, Mr Grayling suggested the US president "perhaps doesn't understand" the shift in power that has taken place between the UK and Brussels.
"I can only think he doesn't realise," he added.
Mr Grayling also insisted that claims by French economy minister Emmanuel Macron that his country would end an agreement that effectively moves Britain's border over the Channel were a "red herring".
He told Marr: "This is a great red herring. Let's be clear, the French government, the French interior minister, have said very clearly that won't change."
Mr Grayling said Eurostar would be told to check before passengers boarded that they had a right to come to Britain.
"Lots of people would turn up in France and discover actually there wasn't a way to Britain and France would have a bigger refugee problem than it does now."
Mr Macron underlined his claims that France could reconsider border arrangements with Britain.
He told the programme: "From a political point of view, it's obvious it will be extremely tough vis a vis our people, especially in the north of France, to explain, OK this guy decided to leave but we will keep everything like it was."
Mr Macron said the EU would have to show there were "consequences" of Brexit in its negotiations with Britain.
"We have to be very clear that Brexit will have consequences, otherwise that is the beginning of the dismantling. Everybody who disagrees on one or two points will decide to do the same.
"On passports, for sure, on access to the single market, because you will be no more part of the club."
Chancellor George Osborne seized on Mr Macron's comments, tweeting that the minister "makes clear" Britain "would have to pay into EU budget and accept free movement to access single market" after Brexit.
Work and Pensions Secretary Stephen Crabb warned that quitting would cause an "economic rupture" similar to the 2008 banking crisis and would have "disastrous" consequences for families.
Cricketing hero Sir Ian Botham, meanwhile, branded the EU a "racket" as he entered the Brexit debate in support of the Leave campaign.
Stepping up to the crease in The Sunday Times, he said it was "insane" that Britain was restricted in its trade with "natural friends" from outside the bloc and labelled some of its programmes, in particular the Common Agricultural Policy, "horrors".
He wrote: "Cricket is a game where you achieve success when you are confident in your own ability to go out and stand proud. Britain has that spirit."