The DUP has welcomed the government’s “renewed focus” on addressing its objections to the Brexit deal ahead of next week’s third Commons vote.
The party has twice voted against the deal over concerns it would see Northern Ireland treated differently from the rest of the UK.
After talks with ministers in London, its Westminster leader Nigel Dodds said it was still seeking extra guarantees.
His party “wanted to get a deal but it had to be the right deal”, he said.
Mr Dodds spent Friday afternoon in meetings with key cabinet figures – including Chancellor Philip Hammond and Environment Secretary Michael Gove – as the government seeks to persuade MPs to support its deal when it returns to the Commons.
The third “meaningful vote” on Mrs May’s deal is expected by 20 March and, if agreed, the prime minister has promised to seek a short extension to the Brexit departure date of 29 March, after MPs voted in favour of a delay.
If it fails to gain support, having already been defeated in the Commons by large margins twice, Mrs May has warned a longer extension may be needed and the UK may have to take part in European elections.
The 10 votes provided by the DUP, which has a parliamentary pact with the Conservatives, are thought to be key to the prime minister securing her deal.
Some Tory Brexiteers who have also criticised the backstop – a fallback arrangement designed to avoid the return of physical checks on the Irish border – and voted against the deal are now pledging their support to avoid a long extension.
James Gray said he will vote for the “obnoxious” deal “after a great deal of soul-searching”, and described those who said they would oppose any deal as “total extremists”.
And former cabinet minister Esther McVey – who resigned her role over Mrs May’s Brexit deal – also suggested she might vote in favour of it.
Some MPs have suggested looking into whether the backstop could be solved by using Article 62 of the Vienna Convention – which would allow the UK to withdraw from any treaty if there had been “a fundamental change of circumstances… which was not foreseen by the parties”.
In a letter to the Times, cross-bench peer and QC Lord Pannick said the UK would be “entitled to terminate the withdrawal agreement” under this clause – although he questioned whether it would be “wise politically”.
Leader of the Commons Andrea Leadsom said the government’s Attorney General, Geoffrey Cox, had considered the matter and would comment further if he thought it was necessary.
Speaking after the meetings in the Cabinet Office, Mr Dodds told reporters there had been “constructive dialogue”.
He added: “Our focus… has been on how can we ensure Northern Ireland leaves the European Union with the rest of United Kingdom as one country.
“We have had good discussions today [and] those discussions will continue over period of time.”
Mr Dodds said his party were “disappointed” with the last minute additions to the deal around the backstop that Mrs May brought back from Strasbourg on Monday night, which she had hoped would persuade MPs to back her plan.
But her Attorney General Mr Cox told Parliament the risk of getting locked into the backstop indefinitely had not changed, and it was later rejected by 149 votes.
Mr Dodds said DUP members were “disappointed” with his assessment and agreed that Mrs May had not made “sufficient progress” around the issue.
But, he added: “We have always said that we want to get a deal, but it has to be the right deal.
“Some of our concerns are not new. What is new now is a renewed focus in government in ensuring those issues are addressed.”
In a week of drama in Westminster, MPs rejected Mrs May’s withdrawal agreement for a second time and then voted against the idea of the UK leaving the EU without a deal under any circumstances.
The Commons then voted to seek an extension to Article 50 – the legal mechanism by which the UK is due to leave the EU.
However, as things stand, the law has not been changed, as Wednesday and Thursday’s votes were not legally binding. That means the UK is still set to leave on 29 March – with or without a deal.
If the government decided it did want to delay, it would have to be agreed by all other 27 EU members. Talks about possible conditions could take place before EU leaders meet at a summit on 21 March.