Chancellor Philip Hammond has refused to back a call from Boris Johnson for £100 million extra a week for the NHS after Brexit.
In a break with normal ministerial protocol, allies of the Foreign Secretary let it be known that he intended to use the weekly meeting of the Cabinet to press for extra health funding.
But the disclosure drew a withering response from Mr Hammond who made clear the NHS was not his department.
"Mr Johnson is the Foreign Secretary," he told reporters as he arrived for a meeting of EU finance ministers in Brussels.
"I gave the Health Secretary an extra £6 billion at the recent Budget and we will look at departmental allocations again at the spending review when that takes place."
Mr Johnson's latest intervention also provoked an angry response from some Tory MPs, with former minister Anna Soubry warning Theresa May he would bring her down unless she sacked him.
The row broke out amid signs of growing frustration among some backbenchers at the Government's performance prompting renewed speculation that the Foreign Secretary was "on manoeuvres".
Last week he again drew attention to his controversial claim during the EU referendum that leaving the bloc would release an additional £350m a week to spend on the NHS, claiming the figure was actually an underestimate.
It is understood he intends 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.
Ms Soubry, a prominent pro-EU campaigner, said the time had come for the Prime Minister to get rid of him.
He had shown "longstanding incompetence and disloyalty" and unless Mrs May acts now "Boris will bring her down", she said.
Mrs May's former chief of staff Nick Timothy also hit out at Mr Johnson.
"Breaching collective responsibility and leaking Cabinet discussions are bad enough but part of political life," he said.
"But pre-briefing your disagreement with Government policy ahead of Cabinet?"
Justice Minister Phillip Lee said "now was not the time" for such a debate, adding: "I'm getting on with my own job - as should others.
Others, however, continued to express concern about a lack of boldness from Mrs 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," he warned.
"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 that it was "time to raise your game" and claimed her Government "constantly disappoints".
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."