Donald Franciszek Tusk is a Polish politician who has been the President of the European Council since 2014. He served as Prime Minister of Poland from 2007 to 2014 and was a co-founder and chairman of the Civic Platform political party.
He said they had had "serious detailed discussions" and things were "moving forward with momentum".
Mr Barnier said it had been a "cordial" meeting, but "lots of work has to be done in the next few days".
The deadline for the UK to exit the EU is 31 October.
And a leaked memo from the European Commission said the UK had confirmed its proposals for replacing the Irish backstop "do not amount to legally operational solutions and would have to be developed during the transitional period".
The memo also said the UK's proposals did not avoid a hard border on The Island of Ireland, or preserve the integrity of the single market.
However, a UK government source insisted their proposals were "serious and workable" and would avoid a hard border.
"Leaks from Brussels on Twitter are par for the course, you can set your watch by them.
Meetings get excitedly talked up in advance (I plead guilty). Anodyne statements follow shortly afterwards.
The UK says it is dreaming up all sorts of ideas.
So What happens next? New York . A shindig at the United Nations next week - and with it the chance for the Prime Minister to have one-on-one chats with the likes of President Macron of France and Chancellor Merkel of Germany.
Could they provide the oomph needed to knock together a Deal ?
It still looks less likely than likely, but who knows?
Mr Johnson, who has said he wants to leave the EU - preferably with a Deal - by 31 October, has urged the EU to scrap the backstop.
But the EU has asked for detailed alternative proposals.
Mr Barclay said this "underscores the purpose there is on both sides to get a Deal and that is what we are working very hard to secure".
He added that "a clear message has been given both by President Juncker and the Prime Minister " and both sides were "working hard".
Also speaking after The Meeting , Mr Barnier said he was not optimistic or pessimistic but "still determined".
He said that any proposal from the UK to replace the backstop "must reach all the objectives of the backstop".
These were to "protect The Peace in Ireland, to protect the all=Ireland economy, and also to protect the consumers and the businesses of the EU and the single market", Mr Barnier said.
The Prime Minister has said the UK needs to leave in a way that allows it to "do things differently" and "not remain under the control of the EU in terms of laws and trade policy".
On Thursday, the UK government said.
This happened after Finland's Prime Minister said Mr Johnson had 12 days to set out his Brexit plans to the EU - although a government source said the development was not in response to those remarks.
Its president, Lady Hale , promised a decision early next week.