Video report by ITV News Europe Editor James Mates
Theresa May held "robust but constructive" talks with European Commission president Jean-Claude Juncker as she sought changes to her Brexit deal.
Mr Juncker underlined that the Withdrawal Agreement thrashed out between the EU and Mrs May would not be redrawn, although he held open the possibility of adding "more ambitious” wording to a document setting out plans for the future relationship.
But Mrs May said: "I am clear that I am going to deliver Brexit. I am going to deliver it on time.
"That is what I am going to do for the British public, I will be negotiating hard in the coming days to do just that."
Video report by ITV News Political Correspondent Libby Wiener
Mrs May is meeting EU leaders, in the wake of the latest war of words between the two sides.
The spat was triggered by European Council president Donald Tusk saying there was a "special place in hell" for those who pushed for Brexit without a plan.
The Prime Minister said she raised with Donald Tusk "the language that he used yesterday, was not helpful and caused widespread dismay in the United Kingdom".
Mrs May said she had told Mr Tusk they should be working together to reach an agreement on Brexit.
"The point I made to him is that we should both be working to ensure we can deliver a close relationship between the United Kingdom and the European Union in future and that is what he should be focusing on," she added.
Following talks with Mr Juncker, a joint statement said: "The discussion was robust but constructive.
"Despite the challenges, the two leaders agreed that their teams should hold talks as to 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."
Brexit Secretary Stephen Barclay and EU chief negotiator Michel Barnier will meet next Monday, while Mr Juncker and Mrs May will have another meeting before the end of the month to take stock of the situation.
Mrs May set out MPs’ demands for a "legally binding change to the terms of the backstop", while Mr Juncker "underlined that the EU27 will not reopen the Withdrawal Agreement."
But he "expressed his openness to add wording to the Political Declaration” to be "more ambitious in terms of content and speed when it comes to the future relationship."
As Mrs May arrived in Brussels a protester waving a placard reading "Don’t crash out" leaped in front of the PM’s convoy in Brussels.
In Westminster, [Labour leader Jeremy Corbyn wrote to the PM setting out five demands](http://Jeremy Corbyn faces backlash over Brexit offer to PM), including joining a customs union, which would need to be met for Labour to back the Government on Brexit.
Mr Corbyn's demands are:
A permanent customs union to deliver frictionless trade and help avoid a hard border on the island of Ireland
Close alignment with the single market, including shared institutions and obligations
Alignment on rights and protections so that UK standards keep pace with those across Europe as a minimum
Participation in EU agencies and funding programmes
Unambiguous agreements on the detail of future security arrangements, including access to the European Arrest Warrant and vital shared databases
Downing Street said ministers were "with interest" at Mr Corbyn's proposals and "welcomed" the fact that he wished to "engage", but added that "there are obviously very considerable points of difference that exist between us.
"The PM continues to believe that an independent trade policy is one of the key advantages of Brexit.
"Her position on the customs union hasn't changed."
Mrs May has promised to take the UK out of the customs union
At the heart of the Brexit crisis is the backstop, which is intended to prevent the return of a hard border in Ireland, would see the UK continue to obey EU customs rules after a transition period if no wider trade deal had been reached.
Mrs May was offered a chink of light by German Chancellor Angela Merkel, who again ruled out reopening the Withdrawal Agreement, but said she believed "solutions" could be found.
Speaking during a visit to Slovakia, Mrs Merkel said: "I think we can find solutions without reopening the Withdrawal Agreement. That is not on the agenda for us."
The possibility of a "disruptive" Brexit is causing economic uncertainty in the EU, commission vice-president Valdis Dombrovskis said, as Brussels revealed its latest economic forecast.
The EU economy is expected to grow 1.5% in 2019, with the UK’s forecast at 1.3% - although this is a "purely technical assumption” based on a status quo relationship with Brussels.
Mr Dombrovskis said: "The possibility of a disruptive Brexit creates additional uncertainty."
Where did the word Brexit come from?