An embattled Theresa May has ordered senior cabinet ministers to go on mini European tours during parliament’s summer break to promote her controversial Chequers agreement on Brexit, as opposition to the deal appears to be hardening in EU capitals and among grassroots Tory party members at home.

The prime minister has warned that “the clock is ticking”, and that the pace of negotiations must be stepped up if a deal is to be reached by the autumn that will benefit both UK and European citizens.

Downing Street is aware that the “hard sell” approach is a high-risk one, and that if the diplomatic offensive fails, and the deal is rejected by the EU, the prime minister’s credibility will have taken yet another blow.

A detailed timetable of summer visits by cabinet members to European capitals has been drawn up by No 10 in the knowledge that only about four weeks of full negotiating time will be left to reach a Brexit agreement once the political holiday season ends.

With many MPs resigned to the prospect of a “no deal” outcome, May will take the lead with a whistle-stop tour this week during which she will meet the Austrian chancellor, Sebastian Kurz, the Czech prime minister, Andrej Babiš, and the Estonian prime minister, Juri Ratas.

Later in July the chancellor of the Duchy of Lancaster, David Lidington, will visit Paris, and the foreign secretary, Jeremy Hunt, will travel to Germany. The home secretary, Sajid Javid, will go to Spain, and the business secretary, Greg Clark, to Italy.

These visits will be complemented by a raft of other ministerial activity across the EU, and by visits from European ministers to London during what promises to be a summer of unrelenting Brexit activity.

With a note of urgency, the prime minister, who appears to have seen off the immediate threat of a leadership challenge after surviving another week of knife-edge parliamentary votes, said on Saturday night: “We must step up the pace of negotiations and get on to deliver a good deal that will bring greater prosperity and security to both British and European citizens. We know the clock is ticking – let’s get on with it.”

She added: “The white paper sets out in detail our proposal for an unprecedented economic and security partnership – it is the right approach both for the UK and the EU.

“Over the summer, our negotiating team will be travelling to Brussels to negotiate our vision for the future relationship with our EU partners.”

May’s problems in selling the deal at home are highlighted, however, by a new survey of grassroots Tories, carried out by ConservativeHome, the website for party activists.

Among 1,350 Tory activists who responded to a survey, 68% said that they thought the Chequers plan would be bad for the country. Two weeks ago, in the immediate aftermath of the Chequers deal, 60% disapproved. The further fall suggests that the May’s attempts to rally support behind it have achieved nothing.

Ominously for the prime minister, Tory sources said there was some evidence in constituency parties that former members of Ukip were joining the party in order to have a vote in any future leadership contest.

Under the Chequers deal, which led to the resignations of the then Brexit secretary, David Davis, and the foreign secretary, Boris Johnson, followed by several other more junior figures, the UK would retain close links with, and access to, the EU single market for goods by following a common rulebook with the EU.

On Friday, the EU’s chief negotiator, Michel Barnier, said the “facilitated customs arrangement” raised practical, legal, economic and budgetary questions.

Referring to his first session of talks with the new Brexit secretary, Dominic Raab, last Thursday, Barnier said he was concerned that European businesses would face higher administrative costs.