Theresa May has sought to calm fractious Tory grassroots activists angry with her Chequers Brexit plan, writing individual members a lengthy letter endorsed by sceptical cabinet Brexiters including Andrea Leadsom and Liam Fox.

The prime minister’s charm offensive over the past few weeks has been aimed at convincing Conservative party members of the merits of her offer to the EU, outlined in the Brexit white paper. It includes a free-trade area for goods governed by a common rulebook.

The deal had prompted David Davis and Boris Johnson to quit the government, saying it was not in line with their vision of a post-EU Britain.

The letter from May, received by Tory party members on Tuesday, follows a concerted effort by Conservatives to woo party chairs. There have been in-person briefings with May’s chief of staff, Gavin Barwell, and the party chair, Brandon Lewis.

Tory chiefs have been alarmed at the extent of the backlash from the grassroots and May herself took part in a conference call to take questions directly from local chairs.

Letter from PM 'selling' her Chequers deal to party members has arrived: pic.twitter.com/FrOjQgVmPi — Phil King (@CllrPhilKing) August 8, 2018

In the three-page letter, estimated by ConservativeHome to have cost the party about £70,000, May repeatedly acknowledges the grassroots’ disquiet, referring to the “strong feelings members of our party have on this important national issue”, and saying there had been “a number of questions” about how a common rulebook would affect sovereignty.

The letter takes aim at critics who have described the Chequers’ offer as capitulation. “It is in no sense a concession to their demands,” she wrote. “I have been very clear that we are rejecting the two models they have put forward.”

Polling by the ConservativeHome website found members bitterly opposed to the Chequers proposal, three in five calling it a bad deal.

A further survey by the site discovered that the popularity of cabinet ministers who backed the deal – including former leadership frontrunners Michael Gove and Sajid Javid – had plummeted. Johnson, however, soared to the top of members’ preferences for next Tory leader for the first time since 2016.

On Wednesday a handful of party members posted pictures on social media of themselves discarding May’s letter or tearing it into pieces.

Members were also sent a link to a script from CCHQ defending the Chequers deal, which had the slogan: “Over the last two years we have examined every possible vision of Brexit. This is the right Brexit for Britain.”

The script preps Tory members to answer questions such as “is this Brexit in name only?” and “won’t the EU just reject this?” which includes the reply “both sides have to compromise”.

Among those endorsing the deal on paper are the Eurosceptic cabinet ministers Fox and Leadsom, as well as Gove, who is environment secretary, and the Brexit secretary, Dominic Raab.

Leadsom, who was reported to have told colleagues she hated the Chequers deal but was prepared to back the prime minister, is quoted as saying: “This proposal puts the power back in the hands of parliament ... this allows our laws to be decided directly by those we elect.”

Fox’s endorsement says the deal will “give the UK the freedom to sign our own independent trade deals”. Raab calls it a “credible vision ... a principled and practical Brexit”.

May’s letter also takes pains to emphasise future trade. “I know that some people are concerned that this common rulebook will stop us doing trade deals,” she wrote. “I can assure you this is not the case – and I would not be proposing it if it would.”

Among those endorsing the deal was Rob Semple, chair of the National Conservative Convention, the members’ body. In his quote Semple appears to acknowledge some differences in the party over the plan, saying it took “on board differing views within the Conservative party and country as a whole ... it is therefore incumbent on the party that we support the prime minister”.