The home secretary, Sajid Javid, has demanded more than £1bn extra funds for police and border force officials to cope with a no-deal Brexit in a request met with scepticism by the chancellor, Philip Hammond.

In a sign the costs of an unresolved Brexit situation are mounting across the board, cabinet members clashed on Tuesday over whether it was necessary to plough more money into preparations for the possibility of leaving without a deal in October.

Javid requested the money to pay for police and border staff who were hired in large numbers before the last March deadline and have been kept on in case they are needed later in the year.

Almost all of the Conservative leadership contestants have, with varying degrees of enthusiasm, insisted that a no-deal Brexit must remain a possibility.

There was, however, no agreement at cabinet to step up any spending on a no-deal scenario, despite demands from Stephen Barclay, the Brexit secretary, to bolster preparations.

It is understood Javid has now formally written to the Treasury asking for between £1bn and £2bn in extra funding for the Home Office.

Despite the promises of some candidates for the Conservative leadership to take the UK out of the EU at the end of October, it is unlikely Hammond will look on Javid’s demand favourably. The Treasury allocated at least £500m to the department when £2bn of Brexit preparation spending was shared across Whitehall.

A source with knowledge of the meeting said: “The home secretary raised the need for additional police and border force no-deal funding. The chancellor’s response and Liz Truss’s was that we’ve not had this bid in and it would be looked at if there is a genuine need, but it would be quite surprising given the Treasury has already released Brexit funding – of which the Home Office got the largest part – so they shouldn’t have run out by now.”

Boris Johnson accused of cowardice as he dodges public scrutiny Read more

Amber Rudd, the work and pensions secretary, who is supporting Jeremy Hunt’s leadership bid, was among those understood to have argued against increasing spending on the possibility of a no-deal Brexit, saying public services should be a greater priority.

Theresa May also told the leadership contenders who are prepared to allow a no-deal exit that parliament is unlikely to stand for it.

She told cabinet her successor would have the same problem that she did – that there were as many Conservative MPs willing to stop a no deal as there were colleagues who would vote down her deal.

During the lengthy meeting, tensions also rose when the attorney general, Geoffrey Cox, said it was not illegal to prorogue parliament to force through a no-deal Brexit, citing precedent. He did, however, stress that it would be unconstitutional and inadvisable to do so.

The extent of no-deal preparations has been a controversy for months after the UK failed to leave the EU at the end of March. May stood down Operation Yellowhammer, the government’s contingency planning operation for dealing with the worst-case scenarios resulting from a no-deal Brexit, after that point.

Staff who had been seconded from elsewhere were returned to their normal duties, but there was no clear role for an estimated 4,500 new recruits. About 16,000 civil servants have been redeployed or recruited to fill Brexit-related posts.

May later wrote to civil servants to tell them that some no-deal preparations would have to carry on, and that plans for crashing out of the EU were necessary and would continue to be signed off by permanent secretaries.

Barclay has been a vocal supporter in cabinet of continuing to spend money on no-deal preparations, but others have argued that is wasteful given that parliament will not let it happen.