Video report by ITV News Europe Editor James Mates
Theresa May and European Commission President Jean-Claude Juncker said they are "seized of the tight timescale" facing them after holding "constructive" Brexit talks.
The comments came after the prime minister travelled to Brussels to seek legal assurances on the Northern Ireland backstop she believes are needed to secure parliamentary approval for her Withdrawal Deal.
Following the meeting, May said that she has "underlined the need for [the UK] to see legally binding changes to the backstop," that would allow it to pass the House of Commons.
A statement, released on behalf of both sides of the negotiations, said that both parties will continue to investigate options for the crucial negotiating point.
It continued: "They will review progress again in the coming days, seized of the tight timescale and the historic significance of setting the EU and the UK on a path to a deep and unique future partnership."
The statement said discussions had looked at "which guarantees could be given with regard to the backstop that underline once again its temporary nature and give the appropriate legal assurance to both sides."
Talks also covered "the role alternative arrangements could play in replacing the backstop in future."
The statement added: "Both reconfirmed their commitment to avoiding a hard border on the island of Ireland and to respect the integrity of the EU’s internal market and of the United Kingdom."
The two leaders said they tasked the EU’s chief negotiator Michel Barnier and Brexit Secretary Stephen Barclay "with considering the process the European Commission and the UK will follow."
Th discussions looked at whether changes can be made to the Political Declaration on the future relationship between the UK and EU that are consistent with positions taken by both sides.
The PM and Mr Juncker agreed to talk again before the end of the month.
Mr Juncker said before the meeting that he did not expect a “breakthrough” at the talks with Mrs May.
The PM believes that gaining legally binding assurances that the backstop will not extend indefinitely is the key to winning the support of MPs for her deal, and seeing off efforts to extend the negotiation period in a series of Commons votes expected on February 27.
The backstop arrangements would see the whole of the UK remain in a customs union with the EU and Northern Ireland following some single market rules until a wider trade deal is agreed, in order to prevent the need for checkpoints on the Irish border.