THERESA May defended the Brexit deal struck at Chequers as Downing Street stepped up efforts to secure Tory Eurosceptic support for a plan that will keep the UK closely tied to Brussels.

The Prime Minister said the proposals thrashed out with the Cabinet were aimed at making sure "we deliver on Brexit for the people because I won't let people down".

May insisted that the UK would "regain control of its borders", but refused to rule out giving EU citizens preferential treatment under the future immigration policy.

The Prime Minister's inner circle is trying to secure support for the proposals to create a new UK-EU free trade area for goods, with a "common rulebook".

Cabinet Brexiteers have already been persuaded to support the plans after the marathon session of talks at Chequers on Friday, but backbench Eurosceptics have expressed grave doubts.

Full details will be published in a white paper on Thursday, with Brussels expecting a round of negotiations in the week commencing July 16.

The Prime Minister said: "We're leaving the European Union. I think when people voted to leave the European Union, they wanted an end to free movement - free movement will end.

"They wanted us to end the jurisdiction of the European Court of Justice in the UK – that will end.

"They wanted us to stop sending the vast sums of money every year to the EU that we do today, and so take control of our money, our laws and our borders – and that's exactly what we will do.

"But we'll do it in a way that protects drops and enhances our economy for the future."

Repeatedly pressed on whether the new immigration system would offer preferential access to EU citizens, May said: "We need to look at that in the context of the wider rules we have for immigration from outside of the European Union and we will decide the rules that are right for the UK."

The Prime Minister said that "collective responsibility has returned" to the Cabinet after the Chequers deal and it was now important to get on with negotiating with the EU.

"Up till now what we've seen from Europe are the proposals that they've effectively put to us of being ones that we couldn't accept," she said.

She said there was a "willingness to sit down and talk" about the plans.

The EU's chief Brexit negotiator Michel Barnier said he was looking forward to the publication of the white paper and the EU would consider whether the proposals are "workable and realistic".

As well as potentially difficult talks ahead with Brussels, May also faces the task of winning over Eurosceptic backbenchers.

Jacob Rees-Mogg, leader of the influential European Research Group of pro-Brexit Conservative MPs, warned that a common rulebook could make "trade deals almost impossible" if it meant regulations would have to apply to any goods coming into the UK.

He added that "it is possible that this deal is worse" than a "no deal" Brexit.

"An egg that is very softly boiled isn't boiled at all. A very soft Brexit means that we haven't left, we are simply a rule-taker," he said.

"That is not something that this country voted for, it is not what the Prime Minister promised."

Backbencher Andrea Jenkyns said she was "awaiting the detail" of the plans before deciding whether to support calls for a leadership contest.

Commenting on the proposals, Stephen Gethins, the SNP's Foreign Affairs and Europe spokesperson at Westminster, said: "This isn't an agreement – it is another helping of fudge served by a UK government that is immobilised by incompetence and devoid of any real sense of direction.

"Two years on from the EU referendum and the UK government is still negotiating with itself rather than with the EU, and what is being proposed is likely to be rejected by the EU when the full details emerge in the long overdue Brexit white paper. 

"The UK government's very own economic analysis warns of the dangers that a no-deal Brexit will have on Scotland and the UK. The evidence has been on her table for months on end, yet the Prime Minister is more intent on appeasing the hard Brexiteers in the Tory ranks.

"The Scottish Government has once again published a detailed compromise plan over its preferred position that looks to protect the economic and security relationship, labour market, and the living standards of families. It has been shared with Theresa May and she must end her disdain for the devolved administrations and acknowledge its findings."

Tories are being invited to briefings about the plans, with the Prime Minister set to address Conservative MPs at a meeting in Parliament on Monday.

The first briefing was led by Chief Whip Julian Smith and the Prime Minister's chief of staff Gavin Barwell, with 10 Tory MPs attending the Saturday morning session.

James Cleverly, a deputy chairman of the party, said: "I went in there with some concerns as a Brexiteer and I come out with those concerns addressed."

He said the briefings "will massively calm the nerves of people who have been basing their views on the speculation that has been floating around in the media and social media".

Vicky Ford urged colleagues not to get "too hot under the collar" until they had seen the details.

Transport Secretary Chris Grayling acknowledged the plans could cause "issues" in future trade talks.

Asked what the common rulebook with the EU on food products would mean for the proposed US trade deal, Grayling said: "There may be individual issues to address in future trade talks."

Taoiseach Leo Varadkar spoke to May about the plans on Saturday.

He said he "welcomed proposals from British Government which can input into talks on the future relationship".

"Await detail in white paper late next week before agreeing an EU position," he said.