Theresa May will return to Brussels on Wednesday to continue talks on her Brexit deal. She will meet the President of the European Commission, Jean-Claude Juncker. Earlier, Brexit Secretary Stephen Barclay updated Cabinet on talks with the EU's chief negotiator, Michel Barnier.
The meeting, on the issue of the Irish backstop, was described as productive but Mr Barnier expressed concerns.
Jeremy Corbyn also announced he would be going to Brussels to meet Mr Barnier on Thursday.
The Labour leader said they would discuss his party's Brexit proposals - including a permanent customs union and a strong relationship with the single market - and that it was a necessity to take no deal off the table.
The backstop is the insurance policy for stopping a hard border returning to the island of Ireland after the UK leaves the EU on 29 March.
The policy, which is part of Mrs May's withdrawal agreement from the bloc, became one of the main reasons her deal was voted down in Parliament in January, as critics fear it would leave the UK tied to a customs union with the EU indefinitely.
MPs gave their backing for Mrs May to renegotiate the policy with the EU in a vote earlier this month. The prime minister said she was working hard to secure the legally binding changes that Parliament wants.
Barclay and Attorney General Geoffrey Cox met Mr Barnier on Monday to discuss ongoing work in the UK with the Alternative Arrangements Working Group - a selection of Conservative MPs trying to find something to replace the backstop.
The group was set up following signs of support for the so-called Malthouse Compromise - proposed by Remainers and Leavers - to extend the transition period for a year until the end of 2021 and to protect EU citizens' rights, instead of using the backstop.
But a spokesman for the Department for Exiting the European Union said: While the [European] Commission engaged seriously with these proposals, it expressed concerns about their viability to resolve the backstop.
He added: We agreed to keep exploring the use of alternative arrangements - especially how they might be developed to ensure the absence of a hard border in Northern Ireland on a permanent footing, avoiding the need for the backstop to ever enter force.
Mrs May and Mr Cox - who will also head to Brussels on Wednesday - have also been looking at other options to replace the controversial backstop, including putting a time limit on it or ensuring the UK can leave the arrangement when it chooses.
But senior figures in the EU have consistently said they will not re-open the withdrawal agreement or scrap the backstop. On Tuesday, a European Commission spokesman said: The EU27 will not reopen the withdrawal agreement.
We cannot accept a time limit to the backstop or a unilateral exit clause - and further talks will be held this week to see whether a way through can be found that would gain the broadest possible support in the UK parliament and respect the guidelines agreed by the European Council.”
The PM has promised to return to Parliament to update MPs again on 26 February and, if she had not got a new deal by then, to give them a say on the next steps in non-binding votes.