UK justice secretary Michael Gove, who supports Brexit, has asked the government to pledge the money in the event of an EU exit saying it could be funded through the UK's EU budget.
Gove's call came at a televised Q&A, and called on voters to "take back control" from "Europe's elites".
Prime Minister David Cameron said the Leave campaign was writing cheques it knew would bounce. "Nine out of 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."
The UK's EU referendum will be held on 23 June.
The Leave campaign's funding proposal came in a statement from Gove, Conservative MP Boris Johnson and Labour's Gisela Stuart, who chaired the Leave campaign.
They said, "As our population grows, and as we all live for longer, so the pressures on the NHS are set to grow.
"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."
If the country votes to leave on 23 June, they said, "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."
Meanwhile, Gove hit out at the ''invincible arrogance of Europe's elites'' in an interview on Sky News, in which he accused the prime minister of scaremongering.
Lashing out David Cameron, Gove told a studio audience that Remain supporters were treating UK voters as if they were ''too small, too poor and … too stupid'' to go it alone outside the EU.
Speaking about Cameron's appearance on Thursday night, Gove said, ''I believe that last night, I am afraid, what we heard was depressing and it was an exercise in trying to scare you into not following through what you know in your heart to be right.''
He said his opponents were guilty of a ''depressing litany of projections about world war three and global Brexit recession,'' saying it was the leave campaign that was ''project hope''.