Tory MPs backing a hard Brexit have fired a warning shot at Theresa May over her Brussels breakthrough deal – one saying it must be “refined”.

No hardliners came out in direct opposition to the agreement, but several made clear their fears that it would prevent Britain signing free trade deals with other countries.

The concerns centre on the insertion of the phrase “alignment” with current and future EU regulations – across the UK – if necessary to avoid a hard Irish border.

Many Conservative MPs believe successful trade deals with non-EU countries will involve agreeing different regulations, on food and manufactured goods for example.

David Jones, the former Brexit minister, was most critical, suggesting the deal had to be “refined” when the Prime Minister went back to Brussels for further negotiations.

“The worry about that, of course, is that that could well relate to very important areas such as for example agriculture, which we would want to throw into the mix in negotiating a free trade agreement with a third country,” he said.

“And if this were to persist then it could severely handicap our ability to enter into those free trade agreements.”

Mr Jones also suggested the “divorce bill” could be “considerably” higher than the £39bn put forward by No 10 – describing it as a “back of an envelope” figure.

Another former Cabinet minister, Owen Paterson, described the agreement as a “big improvement” on the text floated, and vetoed by the Democratic Unionist Party, on Monday.

But he tweeted: “Still problematic areas inc Clause 49 (full alignment), role of ECJ & money. At last we are moving to free trade discussions.

“Problem areas must be debated & resolved. Nothing is agreed until everything is agreed.”

The message was echoed by John Redwood, a third former Cabinet minister, who insisted the deal was “not the Agreement on the UK leaving the EU”.

He tweeted: “It is an agreement to talk about all matters, and is still governed by the crucial principle that nothing is agreed until everything is agreed.”

And he added: “A good deal has to be better than this, otherwise the Government’s mantra that No Deal is better than a bad deal should apply.”

Eyebrows were also raised by a tweet by Boris Johnson, even though the Foreign Secretary send his “congratulations to PM for her determination in getting today’s deal”.

Mr Johnson said: “We now aim to forge a deep and special partnership with our European friends and allies while remaining true to the referendum result – taking back control of our laws, money and borders for the whole of the UK.”

In Dublin, Stephen Donnelly, Fianna Fail’s Brexit spokesman, said he feared it showed “influential Brexiteers” interpreting the impact of alignment very differently to the Irish government.

The crucial text read: “In the absence of agreed solutions, the United Kingdom will maintain full alignment with those rules of the Internal Market and the Customs Union which, now or in the future, support North-South cooperation, the all island economy and the protection of the 1998 Agreement.”

The Irish government insisted that meant the UK had agreed to adhere to any trading regulations necessary to avoid a hard border.