The PM will hold talks with the German Chancellor and French President later, four days before the UK is due to leave the EU.
The meetings with Angela Merkel in Berlin, followed by Emmanuel Macron in Paris, are being held a day ahead of an emergency EU summit on Brexit.
It comes after Parliament passed a bill which aims to force Mrs May to request a Brexit extension.
Cross-party talks aimed at breaking the impasse are also set to continue.
The UK is currently due to leave the EU at 23:00 BST on Friday.
So far, MPs have rejected the withdrawal agreement Theresa May reached with other European leaders last year.
On Monday evening, Parliament passed a bill brought by Labour MP Yvette Cooper, which aims to force the prime minister to request a Brexit extension rather than leave the EU without a deal.
However, the final decision on an extension lies with the EU.
The bill received its Royal Assent from the Queen on Monday night, and Commons Leader Andrea Leadsom told MPs that this meant there will be a government motion on Tuesday asking the House to approve the PM’s request to the EU to delay Brexit until 30 June.
Number 10 said ministers and their shadow counterparts will continue cross-party talks later, as they try to break the Brexit deadlock.
A Downing Street spokesman said the government was “committed to finding a way through” which requires both sides “to work at a pace”.
Talks between Labour and the government began last week, with Mrs May saying only a cross-party pact would see MPs agree a deal in Parliament.
On Monday, sources indicated the PM had not accepted Labour’s customs union demand, but there was a move towards changing the non-binding political declaration.
And the government reportedly suggested offering Labour a guarantee that any deal they reached could not be undone, creating a “lock”.
This aims to ease Labour concerns that any promises could be unpicked by the next Conservative leader.
But BBC political editor Laura Kuenssberg said there was “deep concern” on the Labour side that any legal promise could be undone by further legislation.
Labour leader Jeremy Corbyn said there had been no change in the government’s “red lines”.
However, the prime minister has been warned by members of the 1922 committee of Conservative backbenchers that agreeing a customs union with the EU in Brexit talks would be “unacceptable”.
The MPs met Mrs May in Downing Street on Monday and it is understood they were more open to the idea of a customs arrangement, which would allow the UK to do its own trade deals.
If no compromise can be reached between the parties, Mrs May has committed to putting a series of Brexit options to the Commons and being bound by the result.
Mr Corbyn said: “Talks have to mean a movement and so far there’s been no change in those red lines.”
The Labour leader said there were “many concerns” his party had over the political declaration – a plan for the future relationship with the EU – which it planned to put to the government in their discussions.
Meanwhile, the government has taken steps to ensure the UK can take part in European Parliament elections on 23 May.
A Day of Poll Order has been laid in Parliament, which is required by law for the vote to take place.
The Cabinet Office said it was taking responsible steps, but the move did not make participation in the elections inevitable.
Key dates in the week ahead
- Tuesday: Theresa May travels to Berlin for talks with German Chancellor Angela Merkel and then Paris for discussions with French President Emmanuel Macron. Cross-party talks continue
- Wednesday: Emergency summit of EU leaders to consider UK request for further extension until 30 June, with the option of an earlier Brexit day if a deal can be agreed
- Friday: Brexit day, if UK is not granted a further delay