Boris Johnson will have to embark on a whistlestop charm offensive meeting EU leaders over the summer to try to secure an emergency Brexit summit in September if he is serious about avoiding no deal, according to Whitehall sources who have been involved in negotiations.

The senior sources said Johnson, if he becomes prime minister, would immediately need to pull out the stops over parliament’s summer holiday with a round of diplomacy with senior politicians in Europe. He may even need to consider recalling parliament early if he is to achieve the new deal he claims to want.

At the same time, they said, he would need to ramp up preparations for a no-deal Brexit, with small businesses currently complacent about the risks of this outcome.

One senior source who has been involved in talks with Brussels said there were signs there could be some willingness to compromise, but Johnson could not afford to wait until a planned summit in October, weeks before the UK is due to leave.

Another source – a political ally of Johnson – said it would need to be established “extremely quickly” over the summer whether there was enough appetite among European leaders for finding a solution that could satisfy both Dublin and London.

Johnson’s campaign played down the need for such a course of action, but the Tory hopeful has not yet publicly set out how he hopes to achieve a Brexit deal in his first 100 days in No 10.

He has claimed that the chances of a no-deal Brexit are “a million to one”, but his only public strategy so far is to threaten to withhold the £39bn that the UK has agreed to pay the EU in order to secure a new agreement.

Stephen Barclay, the Brexit secretary, will head to Brussels on Tuesday for meetings with Michel Barnier, the EU’s chief Brexit negotiator, to scope out the areas where there might be a willingness to compromise under a new prime minister.

Barclay, an early supporter of Johnson for the top job, has repeatedly argued in cabinet in recent weeks that spending on no-deal preparations needed to be increased.

Johnson’s team favour leaving the EU on 31 October with some sort of standstill agreement during which a new deal and trading relationship can be negotiated. He would be willing to continue paying into the EU budget during this period.

However, this has repeatedly been rejected by Brussels, with the European commission saying Theresa May’s Brexit deal is the only one on offer and that there will be no transitional period if there is no deal.

Meanwhile, the chancellor, Philip Hammond, is pushing for Conservatives to be given a free vote on a parliamentary move to stop Johnson as prime minister suspending parliament to force through a no-deal Brexit.

Hammond is trying to persuade May that she should lift the whip on certain amendments aimed at preventing the House of Commons being denied a say on whether the UK leaves on 31 October without a deal.

Dominic Grieve, a Conservative MP and former attorney general, is leading a cross-party effort to prevent a no-deal Brexit with amendments to the Northern Ireland bill, but it is not certain of passing if the government whips against it.

Hammond’s best leverage is that he still holds the purse strings as May seeks £27bn of extra education spending as part of her legacy before leaving Downing Street in a fortnight’s time.

It is understood that discussions are ongoing over the education spending, and Hammond may be willing to sign off £5bn for the next year, but he believes May’s successor should make the decision about levels of cash for schools after that.

The Times reported that he was making spending on education conditional on May allowing a free vote on the prorogation amendments.

Grieve, supported by opposition parties and a handful of pro-EU Conservatives, laid an amendment to Northern Ireland legislation on Monday that would stop parliament being prorogued in the autumn to achieve a no-deal Brexit.

He told Sky News he hoped to “persuade my colleagues that I think this bill provides a vehicle by which we can ensure the prorogation of parliament can’t happen”.

He said: “I think that the chances of a prime minister carrying out a no-deal Brexit if there is a majority of the House of Commons that doesn’t want it are very slim indeed. And removing the possibility of prorogation makes them even slimmer.

“Obviously, if a prime minister can persuade the House of Commons that a no-deal Brexit is a good idea, he’s fully within his rights to do it. But attempts to ratchet no-deal through, against the majority wishes of the House of Commons, should not happen and, in my view, we can put in place the necessary provisions to make sure it doesn’t.”

It is understood that while Johnson and his rival for the premiership, Jeremy Hunt, plus their supporters among Tory MPs, will miss the vote owing to an ITV debate in Manchester the same evening, this will not affect the result, as Labour will offer a pairing system if requested.

A Downing Street spokesman denied speculation that officials would seek to avoid amendments to a bill that otherwise had what he called “a very clear purpose”. He said: “MPs will table amendments. It’s for the Speaker to decide whether or not they’re selected. That’s the proper way of doing things.”