Boris Johnson is expected to use a Tuesday Cabinet meeting to demand £100 million extra a week for the NHS after Brexit, it is understood.
The Foreign Secretary made clear last week he thinks the health service should be "at the very top of the list" when the UK "takes back control" of money which would have been sent to Brussels as a member.
Britain is expected to keep paying into the EU budget until the end of a post-Brexit transition period in around 2021, and Mr Johnson has insisted the Government will save hundreds of millions of pounds afterwards.
He is set to use Tuesday's Cabinet meeting, when Health Secretary Jeremy Hunt will update ministers on the NHS winter crisis, to demand around £5 billion a year extra for the health service, it is understood.
Mr Johnson was rebuked last week after he said the Leave campaign underestimated when it claimed Brexit would allow an extra £350 million to be spent on the NHS weekly.
Leading health expert Professor Sir Michael Marmot said the Brexit campaign's highly controversial claim has already been confirmed "a lie".
Chancellor Philip Hammond used his autumn Budget to commit £2.8 billion extra to the NHS in England.
But the Foreign Secretary's move comes as Tories complain about a lack of boldness from Theresa May's Government amid the electoral threat of Jeremy Corbyn's radical Labour.
Sir Nicholas Soames is the latest Tory backbencher to voice unease about the style of the Prime Minister's administration, after former minister Nick Boles warned of "timidity and lack of ambition".
Using one of the extended hashtags which have become his social media trademark, Sir Nicholas asked: "Where's the bold and brave? So far, it's dull, dull, dull."
"It really won't be enough to get people to vote against The Corbini," warned Sir Nicholas.
"They must have really sound reasons to vote Conservative.
"We really need to get on with this. #wherestheboldandbravesofaritsdulldulldull."
Sir Nicholas's comments came just days after Mr Boles warned the PM it was "time to raise your game" and claimed her Government "constantly disappoints".
The Grantham and Stamford MP tweeted at the weekend: "There is a timidity and lack of ambition about Mrs May's Government which means it constantly disappoints. Time to raise your game, Prime Minister. #worboys #HousingCrisis #NHSfunding #etcetc"
Asked about the concerns expressed by Mr Boles and Sir Nicholas, Mrs May's official spokesman said on Monday: "If you look at action taken by the Government in recent months, you can see it is getting on with the job of building a stronger economy and a fairer society that works for everyone."
He cited cuts in stamp duty for first-time buyers, the launch of a 25-year environment plan and an industrial strategy, plans for an energy price cap, the recently published race disparity audit and the injection of money into health, social care and schools.
"Of course, we know there is more to do and the Prime Minister is committed to getting on and delivering."