They agreed to call a summit if Barnier decides there is "decisive progress" in the talks.
Talks are stalled on the question of how to avoid a policed frontier between Northern Ireland and the Republic of Ireland, without erecting new barriers between Northern Ireland and the rest of Britain.
The prime minister's high-stakes diplomatic mission started with a handshake and a kiss on the cheek from the president of the EU's executive branch, Jean-Claude Juncker.
Stephen Phipson, director-general at the manufacturers' association, EEF, told the Financial Times: "The offer to extend the transition period by the prime minister is a pragmatic and sensible offer which the vast majority of manufacturers would welcome".
Britain says it has not asked for an extension - but didn't rule it out Wednesday.
"The only answer to public disillusionment with how Brexit has not lived up to what they voted for is a People's Vote on the final Brexit deal".
"Such an extension would cost the United Kingdom billions of pounds, yet our fundamental problem with the European Union proposal remains".
Talks on finalising the UK's withdrawal agreement and a political declaration on future trade relations with the EU have been blocked by failure to agree on a backstop to avoid a hard border between Northern Ireland and the Republic in the event that an agreement on future EU-UK relations can not be struck.
Talking about the CFTC possibly blocking European Union banks from accessing U.S. markets, Mr Guersent said: "I don't like it but that is their right".
Tajani, however, also emphasised that "without an agreement on the three points (EU citzens rights in the UK, Brexit divorce bill and the Northern Ireland matter), the European Parliament can not and will not vote in favour of the agreement".
Mr Varadkar said a longer transition period was not a substitute for a concrete agreement over the backstop.
European Union leaders have recently suggested that the transition period, now due to end in December 2020, could be extended by a year to provide more time for a trade deal to be forged that would make the backstop unnecessary.
The EU has also argued against Britain cherrypicking by staying in the single market for years while having no obligations of EU membership. Any deal that is struck will have to be approved by the British and European Parliaments.
The prime minister will be in Brussels until Friday in an attempt to make progress in Brexit negotiations.
The prime minister failed to reassure MPs from the Democratic Unionist Party that there'd be no new regulatory checks between Northern Ireland and the rest of the United Kingdom after Brexit, and received a kicking from numerous pro-EU MPs, including up to 10 MPs who urged May to back another referendum.
Arriving for the summit, she insisted that a deal is achievable and that "now is the time to make it happen".
However, an extension does not in and of itself solve May's problem, which is that she has yet to fully decide, let alone agree, what sort of relationship she wants to secure during that additional transition period.
"The working assumption is not that we will have no deal".
But she said she had found "a very real sense that people want that deal done" among her fellow leaders. Others at the bar managed to capture videos of the four leaders enjoying themselves and, naturally, posted them online.
In a speech in London on Tuesday evening, John Major, a former Conservative Prime Minister, said the Brexit vote was a "colossal misjudgment" and that Brexit-backers sold the country a fantasy and would not be forgiven.