Theresa May is facing a fresh showdown with Eurosceptic Conservative MPs after a cabinet minister suggested she may put her Brexit deal to parliament again without having secured a change to the withdrawal text.

Before talks between May and EU leaders, the culture secretary, Jeremy Wright, said the prime minister’s aim was to solve the Northern Ireland backstop issue but the “mechanism” of the change did not matter.

His words caused alarm among some Eurosceptic Tory MPs, who have repeatedly made clear they would not vote for a deal that did not remove an indefinite Northern Ireland backstop – a clause that could bind the UK into a permanent customs union with the EU.

May has written to every Tory MP reassuring them that she would attempt to eliminate any risk that the backstop is applied indefinitely and urging them to “move beyond what divides us and sacrifice personal preferences for the national interest”. This week, she will speak to every EU leader and meet Jean-Claude Juncker, the European commission president, in an attempt to break the stalemate over the agreement, which Brussels does not want to reopen.

A No 10 source insisted that May would request a reopening of the withdrawal text. Geoffrey Cox, the attorney general, will give a rare speech on Tuesday setting out how the government thinks the backstop problem can be legally solved. Stephen Barclay, the Brexit secretary, will also meet the EU’s chief negotiator, Michel Barnier, on Monday to discuss proposals to replace the Northern Ireland backstop.

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However, Wright, the former attorney general, suggested it may not be necessary to amend the withdrawal agreement, while calling on MPs to give the prime minister space to negotiate.

He told the BBC’s Andrew Marr Show: “I think what’s obvious is that parliament, and I think probably people well beyond parliament, are concerned about the potential indefinite nature of the backstop – that’s what we’ve got to do something about.

“If this is the only way of doing it then that’s the way we will pursue. If there are other ways of doing it that are just as effective that perhaps we haven’t yet explored then we will do that too.”

He continued: “I don’t think it’s the mechanism that matters, it’s the objective: if you can get to a place where the potential longevity of the backstop, the potential that the backstop lasts forever can be adequately dealt with, that’s what we’re all seeking to do.

“That’s what parliament has been very clear that it wants, it will back this deal if we can do something about the backstop.”

Eurosceptic MPs would refuse to support any solution that simply amounted to an addendum or codicil to the EU deal, saying they want the backstop removed from the withdrawal text.

Steve Baker, the former Brexit minister and an influential figure in the European Research Group (ERG), retweeted a message saying Brexiter Tories would not like the sound of Wright’s suggestion.

He has consistently emphasised that the ERG would rather support a no-deal Brexit than one that did not legally remove the backstop.

In leaked WhatsApp messages, he told fellow Conservative MPs that failure to remove the Irish backstop completely could mean things would “just grind towards a party split” and he expressed fears that the prime minister’s negotiations were “a complete waste of time”.

Sir Bill Cash, a veteran Eurosceptic, said No 10 would have to “show there has been a change in the substance – that requires a change in policy and there is no evidence yet to show there has been a change in policy”.

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Boris Johnson also made clear last month: “We can’t have some codicil or letter or joint declaration. We need to go back into the text of the treaty and solve the problem.”

However, Martin Howe QC, a lawyer supported by many Brexiters, said in the Sunday Telegraph that the EU might be able to perform “keyhole surgery” to the withdrawal agreement in a way that would satisfy all sides if it legally overrode the withdrawal text.

“Only a treaty-level clause which confers an unconditional right on the UK to exit the backstop would work,” he wrote. “Preferably this should be inserted into the actual protocol text, but if it is in an addendum or separate document it must be crystal clear that it overrides the protocol text and is not just some kind of interpretation of it.”

Chris Heaton-Harris, a Brexit minister, insisted on Sunday that May was serious about negotiations and would seek to come up with a solution that satisfied everyone in the party.

He told Sky’s Sophy Ridge on Sunday there was a “huge amount of activity” between the UK and Brussels. “The government is not trying to run the clock down. The government is trying to get a negotiated deal with our European partners,” he added.

May will need the support of Eurosceptic Conservatives to get approval for her withdrawal agreement unless she appeals to enough pro-Brexit Labour MPs. The bloc of ERG MPs showed their strength last week by refusing to support a motion endorsing May’s EU strategy previously approved by parliament.