Diverting the UK's contribution to EU coffers would give the straining health service a "cash transfusion", the Justice Secretary and former London mayor have claimed.
But the economic impact of Brexit would mean there is less money to pump into health care, the Prime Minister insisted.
Mr Cameron said: "The Leave campaign is writing cheques they know will bounce, 9/10 economists say there'll be a profound shock if we leave the EU. That means there will be less money - not more. It's also why so many doctors and nurses support remaining in the EU."
Leave campaigners are holding a London rally championing the Brexit cause.
The Justice Secretary made the NHS funding claims just hours after he faced a live television grilling ahead of the June 23 poll.
In a joint statement, Conservatives Mr Gove and Mr Johnson and Labour's Gisela Stuart , said the number of migrants from within the EU arriving in the country last year amounted to a city the size of Newcastle.
"As our population grows, and as we all live for longer, so the pressures on the NHS are set to grow," they said.
"We believe that one of the best ways to protect, and to strengthen, the NHS, for the people of this country is to use some money we currently spend on EU membership to invest in improving healthcare."
"After we Vote Leave on 23 June, the Government should use some of the billions saved from leaving the EU to give at least a £100 million per week cash transfusion to the NHS," they added.
"This money will be over and above the commitment that the Prime Minister rightly made at the last election to an £8 billion real terms increase."
It comes as all six living former Labour leaders came together to issue a plea for the party's supporters to vote Remain in the EU referendum, warning: "If Labour stays at home, Britain leaves."
In an open letter released by Britain Stronger in Europe, Lord (Neil) Kinnock, Dame Margaret Beckett, Tony Blair, Gordon Brown, Harriet Harman and Ed Miliband warned that Labour voters cannot afford to opt out of involvement in what many see as an internal spat between Conservatives.
Labour's traditional working-class supporters have "the most to lose if we leave, but also the most to gain if we remain", said the letter.
Many observers believe Labour voters could hold the key to the outcome of the referendum, with Remain strategists concerned that a low turn-out by the party's supporters on June 23 could hand victory to the Leave camp.
While polls show the party's voters are significantly more likely than Conservatives to back Remain, they are also less likely to take part in the referendum - with a recent ORB survey finding 52% of Labour voters saying they were "certain to vote", against 69% of Tories and 71% of Ukip supporters.
Mr Brown, who is speaking at the Hay-on-Wye Festival, is calling for a "positive, principled and progressive" message to persuade people to back Remain to support jobs, security and fairness.
In the letter, the six wrote: "Europe protects people at work; stimulates jobs and innovation; keeps prices lower; leads global action against climate change; makes us safer against terrorism; and magnifies Britain's voice and values in the world.
"But make no mistake: this would be lost if we leave. Labour communities would face a double threat: the return of recession, led by a Tory government with an emboldened right wing.
"In such circumstances Labour communities would suffer most: from spending cuts, neglect for the needy and a bonfire of workers' rights.
"Those Labour seeks to represent - the hard-working, ambitious majority - have the most to lose if we leave. But also the most to gain if we remain."