Brussels has said it would be ready to accept movement from Theresa May over her "red lines" ahead of crucial Commons votes on her Brexit strategy.
The Prime Minister faces a showdown with MPs next month over her decision to rule out membership of the single market and customs union.
The European Union's chief Brexit negotiator Michel Barnier appeared to offer encouragement to MPs pushing for the Prime Minister to soften her approach by stressing that "the UK can change its mind".
But he expressed frustration at the state of the negotiations, claiming that the UK was playing "hide-and-seek" by refusing to spell out exactly what it wants.
In a strongly worded speech in Lisbon, Mr Barnier stressed that "time is and will remain tight" and called for progress to "speed up" if details of the future relationship are to be agreed before the Brexit date.
"If the UK wishes to modify its red lines, it will have to tell us so," he said, adding: "The sooner the better."
Calling for "clarity", he said: "To negotiate in an effective way, you must know what the other side wants.
"A negotiation can't be a game of hide-and-seek."
And against the backdrop of a bitter row over the potential exclusion of the UK from work on the Galileo satellite project, Mr Barnier said the EU would not be influenced by a "blame game" which sought to put the "negative consequences" of Brexit on Brussels.
After an unnamed EU official claimed the UK's position was a "fantasy" - prompting an angry backlash from Westminster - Mr Barnier stressed the need for realism from Downing Street.
"We too want an ambitious partnership with the UK in the long term. But to get there we need realistic proposals from the UK," he said.
Those proposals must "respect the institutional architecture and integrity" of the EU.
"I can see the temptation of the blame game which would involve unloading the negative consequences of Brexit on to the EU.
"But we will not allow ourselves to be influenced. I will not allow myself to be influenced.
"It is the UK which is leaving the EU. It cannot, in the act of leaving, ask us to change what we are and how we function."
Mr Barnier also highlighted the need to agree on a "robust" governance regime for the withdrawal agreement - with a role for the European Court of Justice (ECJ).
"It is on these three points - a judicial system for settling disputes, the role of the ECJ and the role of the British judges - that we must build a robust system of governance for our withdrawal agreement," he said.
Mr Barnier also said UK calls for a special deal on data regulation pose "real problems".
The UK has called for the Information Commissioner's Office to have an "appropriate ongoing role" on the European Data Protection Board, and for British businesses to be "effectively represented" under the EU's new "One Stop Shop" mechanism for resolving data protection disputes.
Mr Barnier said Brussels is not able to share its "decision-making autonomy" with a third country and the UK "must face up to the reality of Brexit".
The European Union (Withdrawal) Bill is expected back in the House of Commons in June, where the Tories fear possible rebellions over the customs union and single market.
Liberal Democrat leader Sir Vince Cable said there was cross-party co-operation on the measures.
He told the Press Association: "We take the view that Brexit isn't bound to happen, it may not happen - we think there should be a public vote on the final deal - but while the legislation is going through we want to improve it.
"It is very clear that enormous economic damage will be done if we leave the single market and customs union, we don't have to.
"There are a lot of Conservatives and rather more Labour people who are rebelling against their party line on these issues.
"There is a real chance the Government could be defeated."
There had been "a lot of collaboration - it's happened in the House of Lords, it's happened in the House of Commons".
Sir Vince was using a campaign event ahead of the Lewisham East by-election to seek to put pressure on Jeremy Corbyn to throw Labour's weight behind a referendum on the Brexit deal.
In a letter to Mr Corbyn, he said: "The biggest failure of the Labour Party in the 21st century remains the illegal war in Iraq. On this I think you and I might agree.
"Do not allow Labour appeasement of Brexiteers to be another such failure, but this time on your watch."