Tory MPs have lined up to back Rishi Sunak as the defender of fiscal conservativism after Boris Johnson’s threat to sack him, as Treasury sources pushed back against the idea of a carbon tax.

No 10 insisted the prime minister had full confidence in Sunak but his spokesman did not deny Johnson had threatened to demote the chancellor to health secretary amid a row about a leaked letter pushing Johnson to ease Covid restrictions.

Johnson was rumoured to be considering a reshuffle in the autumn although his spokesman would only say there were no imminent plans for a shake-up.

Sunak is backed by the fiscally conservative wing of the Conservatives and is popular with party members, who view him as committed to keeping tight control of day-to-day spending.

There are several areas of tension between No 10 and No 11 Downing Street but the biggest unresolved areas of spending are how to reach the government’s net zero target and fund a new social care system, with both policies delayed until the autumn.

Kwasi Kwarteng, the business secretary, said on Monday morning that a carbon tax was under discussion, telling BBC Radio 4’s Today programme: “You can also go down a carbon tax route, and these policies we’re discussing with the Treasury.”

But Treasury sources pushed back on this idea, suggesting it was no longer considering a carbon border adjustment tax that could put up a levy on carbon-intensive imported goods.

Asked about the possibility of a carbon tax, a Treasury spokesman said: “We’re leading the world in building back better and greener from the pandemic. We were the first major economy to commit to net zero by 2050 and one of the first to phase out petrol and diesel car sales by 2030, and just last week we announced more ambitious carbon emissions targets.

“But this is about policies, not just targets, which is why the prime minister has outlined an ambitious 10-point plan to achieve our environmental goals.”

Under that 10-point plan, the Treasury’s “net zero review” is considering the “choices across our tax, spend, regulatory and other levers to maximise growth opportunities and ensure an equitable balance of contributions across society”. Sources close to Sunak have said he is trying to mitigate the costs of reaching net zero on lower income families.

Ahead of the autumn spending review, Mark Harper, the former Tory chief whip and leader of the lockdown-sceptic Covid Recovery Group, said he would back Sunak making efforts “to get us back on the road to fiscal responsibility”.

“I think we do need to get back to living within our means at some point, reasonably quickly. It also applies to people on my own side and the spending review, I think it is going to be quite tough and involve making some choices.”

Asked whether Johnson should keep Sunak in post, he said: “There is a bit of August in the stories but if anyone were to suggest moving the chancellor I think that would be a mistake. He was absolutely prepared to do what was necessary last year in an emergency but I think he recognises that we have got to reassert some fiscal discipline. And the public is quite [clear] about this too – that you do have to make choices.”

Another Tory MP said Sunak had substantial backing in the party that Johnson would be “unwise to test”. A third backbencher said he thought MPs recognised Sunak’s role in “trying to keep the PM on the straight and narrow” when it came to spending.

A government figure close to the chancellor told the Telegraph at the weekend: “If he demotes him he’s only signing his death warrant. There’s nobody else as good as Rishi.”

Speaking to broadcasters on Monday morning, Kwarteng backed Sunak to stay as chancellor but acknowledged it was in Johnson’s hands saying “that’s up to the PM” and adding: “I think Rishi is doing a great job as chancellor.”

Asked about the difficult relations in Downing Street, he said: “I happen to think the PM and chancellor get on very well generally. They are both very good … there are always disagreements. There is always going to be a slight creative tension between No 10 and No 11. I happen to think this PM and chancellor are working very effectively together.”

Despite fiscally hawkish conservatives expressing worries about the cost of net zero, there is also a large caucus of more than 100 Tory MPs under the Conservative Environmental Network who are championing policies that would help reach the target.