British Prime Minister Theresa May has said that she saw and heard a desire from European Union leaders to ensure Britain leaves the bloc with a deal.
"What I see and hear from leaders is a desire for us to work together to ensure that we can deliver the UK leaving the European Union with a deal," Mrs May said after meeting Commission President Jean-Claude Juncker and Council President Donald Tusk.
"It is not going to be easy but crucially President Juncker and I have agreed that talks will now start to find a way through this, to find a way to get this over the line and to deliver on the concerns that parliament has so we get a majority in parliament," she 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," Mrs May said.
For his part, European Commission President Jean-Claude Juncker told the British prime minister during their "robust but constructive" talks that the EU would not renegotiate the Brexit deal.
"The talks were held in a spirit of working together to achieve the UK's orderly withdrawal from the EU," Mr Juncker’s spokesman Margaritis Schinas told a news conference, adding the bloc was ready to work more on the accompanying declaration of the EU-UK new ties after Brexit.
Mr Juncker said Mrs May presented various options to address her parliament's concerns about the backstop and that the two agreed their teams would work together on "whether a way through can be found."
Mr Juncker and Mrs May would meet again before the end of February, Mr Schinas said.
Their Brexit negotiators - EU's Michel Barnier and UK's Stephen Barclay - will meet on Monday.
Mrs May arrived in Brussels after insisting that the UK must not be "trapped" in a backstop deal.
She is meeting EU leaders in the wake of the latest war of words between the two sides.
Mrs May was greeted by European Commission President Jean-Claude Juncker at EU headquarters.
Both leaders posed for pictures, but did not answer questions.
Mrs May's latest diplomatic offensive comes as British Labour leader Jeremy Corbyn wrote to her setting out five demands, including joining a customs union, that would need to be met for Labour to back the UK government on Brexit.
At present 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.
Downing Street said that Mrs May is "open to different ways" of achieving her objectives on the backstop.
Mrs May is using the meetings to state that Parliament has sent "an unequivocal message that change is required".
She is due to meet Taoiseach Leo Varadkar in Dublin on Friday.
One of her key messages for EU leaders is that the Commons has made it clear it could support the Withdrawal Agreement as long as concerns about the backstop are addressed.
Mrs May is also stressing that Mr Corbyn also has concerns about the backstop, so it is not just an issue for the Tories and their DUP allies.
In Mr Corbyn's letter to the PM, which follows their Brexit meeting last week, he insists that Labour's Brexit demands must be enshrined in the Political Declaration setting out future relations with the EU.
Mr Corbyn said that securing the demands in law is the only way of achieving Labour support and uniting the country.
He calls for a "permanent and comprehensive UK-wide customs union" that gives the UK a say over future trade deals, and close alignment with the single market, underpinned by "shared institutions".
Mr Corbyn also calls for "dynamic alignment on rights and protections" in order that UK standards do not fall behind those of the EU, as well as commitments on participation in EU agencies and funding programmes, and "unambiguous agreements on the detail of future security arrangements, including access to the European Arrest Warrant and vital shared databases".
The letter drew criticism from some pro-EU Labour figures who said it went against Labour's Brexit stance.
Among the prominent EU figures Mrs May is meeting is European Parliament Brexit co-ordinator Guy Verhofstadt who joined in with Mr Tusk's Brexit "hell" analogy.
Mr Verhofstadt tweeted that Lucifer would not welcome such Brexiteers because "after what they did to Britain they would even manage to divide hell".