Summit host Tusk said the evidence of recent months gave EU leaders 'little reason to believe' that British lawmakers will ratify the Brexit withdrawal treaty before May's preferred June 30 departure.
European leaders gathered in Brussels on Wednesday to decide how long a Brexit delay to grant British Prime Minister Theresa May - and under what conditions.
Without a postponement, Britain is due to crash out of the European Union at midnight on Friday under a "no-deal" Brexit that could trigger economic chaos.
May has embarked on a last-ditch battle to postpone Brexit from April 12 to June 30 to arrange an orderly departure - but European leaders are expected to offer her a longer delay of up to a year.
"There are times when you need to give time," EU Council President Donald Tusk tweeted, as he issued summit invitations.
Summit host Tusk said the evidence of recent months gave EU leaders "little reason to believe" that British lawmakers will ratify the Brexit withdrawal treaty before May's preferred June 30 departure.
"In reality, granting such an extension would increase the risk of a rolling series of short extensions and emergency summits, creating new cliff-edge dates," he said, reflecting concern in EU capitals.
"One possibility would be a flexible extension, which would last only as long as necessary and no longer than one year," he said.
According to a draft copy of the summit conclusions that EU leaders were to receive and debate later in the day, they will agree to an extension to allow May time to ratify the withdrawal agreement.
"Such an extension should last only as long as necessary and, in any event, no longer ," the draft reads.
The other 27 EU leaders will thrash out what date to fill in the blanks on Wednesday.
"If the Withdrawal Agreement is ratified by both parties before this date, the withdrawal will take place on the first day of the following month," the draft, seen by AFP says.
If Britain is still in the European Union when EU parliamentary elections begin on May 23, it must take part in the vote.
"If the United Kingdom fails to live up to this obligation, the withdrawal will take place on 1 June 2019," the draft warns.
If an extension is agreed, Brussels will portray it as a concession to Britain, with some members - particularly France - not keen to see the disruptive Brexit drama drag on much longer.
But May insists she still wants to quickly ratify the withdrawal she agreed with EU leaders last November, but which has been rejected by British lawmakers, and to leave before the EU polls.
On Tuesday, May flew to Berlin to meet German Chancellor Angela Merkel, then on to Paris, where an aide to President Emmanuel Macron said France was open to solutions.
"We've never been closed to the idea of finding an alternative solution to 'no deal' within certain limits and not at any price," the Elysee source said.
Discussions in Brussels are to focus on the length of the delay - the French source said a 12-month extension "seems too long" - and arrangements to limit Britain's influence within the EU.
EU members want to ensure that a semi-detached Britain does not seek leverage in Brexit talks by intervening in choosing the next head of the European Commission or the next multi-year EU budget.
Talks with Labour
Britain was originally due to leave the EU on March 29, but Brussels agreed an extension after the British parliament rejected the withdrawal agreement negotiated with May.
The British premier is hoping a new period of extra time, if granted, will enable her to finally get a divorce deal through the legislature.
British MPs have rejected her deal three times, but May is now in talks with the opposition Labour party to try break the deadlock.
These discussions are moving slowly, and EU negotiator Michel Barnier said May must explain what postponement would achieve.
"The length of the extension must be linked to the purpose - what it's for - and that depends on what Mrs May will say to European leaders tomorrow," he said, after meeting EU ministers in Luxembourg.
A "no deal" - in which Britain crashes out of the EU - is still a possibility.
The International Monetary Fund said Tuesday that Britain risks a serious shock if it leaves the EU without an agreement.
And US Treasury Secretary Steven Mnuchin warned American lawmakers: "I think at this point we need to be prepared for a hard Brexit as a very realistic outcome."