Theresa May is meeting European Council president Donald Tusk and is expected to increase the UK's Brexit divorce bill offer.
But her offer - which could rise from about £20bn to £40bn - is conditional on the EU agreeing to begin talks on post-Brexit trade.
The PM is in Brussels for an Eastern Partnership summit, where leaders will discuss moves to protect former Soviet bloc countries from the new threat posed by Russia.
Speaking on her arrival, Mrs May expressed hope that her one-to-one meetings on the sidelines of the gathering could prompt progress on Brexit talks.
She said: "The summit here today is about working with our eastern partners but, of course, I will be having other meetings.
"I will be seeing President Tusk here today, talking about the positive discussions, the positive negotiations we're having, looking ahead to the deep and special partnership I want with the EU.
"These negotiations are continuing. What I'm clear about is that we must step forward together.
"This is for both the UK and EU to move on to the next stage."
As well as attempting to make progress on the divorce bill, the PM also wants to resolve problems over the Irish border and a new move by Spain to force a 'hard Brexit' on Gibraltar.
The Government's strategy for attempting to move forward the Brexit negotiations was thrashed out at a two-hour Cabinet committee meeting in Downing Street on Monday.
At that meeting, the Cabinet's leading Brexiteers - Boris Johnson and Michael Gove - backed the increased cash offer but demanded a pledge on trade talks in return.
Last week, Mr Tusk told the PM the EU needed greater clarity on the terms of Britain's withdrawal - including the financial settlement - by early December.
He said only then would EU leaders give the go-ahead for phase two of Brexit negotiations to start at their next summit on 14-15 December.
To make life harder for the PM, Irish Prime Minister Leo Varadkar is demanding a written guarantee that there will be no return to a hard border between Northern Ireland and the Republic.
He has told Mrs May this is the price of his support for the second phase of the Brexit negotiations beginning and that without it he is prepared to veto the resumption of talks.
Speaking ahead of Mrs May's talks with Mr Tusk, the Prime Minister's spokesman played down suggestions that it was a meeting exclusively to discuss the divorce bill.
"There are a number of issues which I'm sure they will want to discuss - the financial settlement, that will be one of them, also of course Northern Ireland and citizens' rights," he said.
On Gibraltar, after Spain warned the Rock might be excluded from any Brexit transitional arrangement, Number 10 said: "We are seeking to engage with all member states.
"We are determined to get a deal that works for all parts of the UK family. That deal must work for Gibraltar too."
The Brussels summit is being attended by leaders from Armenia, Azerbaijan, Belarus, Georgia, Moldova and Ukraine, who fear they are under threat from Russia and Vladimir Putin.
The PM will set out her commitment for the UK to continue to play a critical role in European security after it leaves the EU.