The UK's offer on Brexit must be acceptable to the Republic of Ireland before the negotiations can move on, the president of the European Council, Donald Tusk, has said.
Mr Tusk was speaking after talks with the Irish prime minister in Dublin on Friday.
He said: "The UK's future lies - in some ways - in Dublin".
The European Union has said "sufficient progress" must be made on the Irish border before negotiations can move on.
"The Irish request is the EU's request," Mr Tusk said.
"I realise that for some British politicians this may be hard to understand.
"But such is the logic behind the fact that Ireland is the EU member while the UK is leaving.
"This is why the key to the UK's future lies - in some ways - in Dublin, at least as long as Brexit negotiations continue."
In a press conference with Taoiseach (Irish prime minister) Leo Varadkar, Mr Tusk said that the UK's decision to leave the EU had created "uncertainty for millions of people".
"The border between Ireland and Northern Ireland is no longer a symbol of division, it is a symbol of cooperation and we cannot allow Brexit to destroy this achievement of the Good Friday Agreement," he said.
By BBC Political Correspondent Adam Fleming
There is a lively debate about whether the Irish government has a veto over the decision - to be taken at the summit of EU leaders on 14 and 15 December - about whether Brexit talks can move to the next phase.
Call it what you like, but now Donald Tusk has told us for sure that the rest of the EU will do what Ireland decides.
There was a put-down for British politicians who may find it "hard to understand" why this is important.
But there was some comfort for the British government: Donald Tusk shares their view that the issue of the border can only be solved when there is more clarity about the UK's future relationship with the EU.
And Mr Tusk ended by saying "the key to the UK's future lies - in some ways - in Dublin." Is this a hint that the Irish government's suggestion that Northern Ireland remain in the EU's single market and customs union is the answer for the whole of the UK?
Or is it just a reminder that Dublin is first among equals among the remaining 27 members of the EU?
"The UK started Brexit and now it is their responsibility to propose a credible commitment to do what is necessary to avoid a hard border.
"As you know, I asked Prime Minister May to put a final offer on the table by the 4th of December so that we can assess whether sufficient progress can be made at the upcoming European Council.
"Let me say very clearly. If the UK offer is unacceptable for Ireland, it will also be unacceptable for the EU."
The taoiseach thanked Mr Tusk for the solidarity demonstrated by all EU partners and called the EU "a family which sticks together".
He said he was optimistic that a deal could be achieved by Monday.
However, he said any UK offer must indicate how a hard border can be avoided and avoid the risk of regulatory divergence.
On Thursday, the DUP's Sammy Wilson said any attempt to "placate Dublin and the EU" could mean a withdrawal of DUP support at Westminster.
He was responding to reports of a possible strategy to deal with the Irish border after Brexit.
The story suggested that British and EU officials could be about to seek separate customs measures for Northern Ireland after the UK leaves the European Union.
The DUP struck a deal with Prime Minister Theresa May's government in June, agreeing to support Tory policies at Westminster, in return for an extra £1bn in government spending for Northern Ireland.