Addressing MPs in the House of Commons on Monday, May had said a deal was "achievable" while sticking to her principles on the Irish border issue.
"I think that could potentially play into the hands of hardline Brexiteers or people that would oppose a deal that Prime Minister May might put to parliament", he added.
"We need much time, much more time and we continue to work in the next weeks", EU Brexit negotiator Michel Barnier said Wednesday, according to The Associated Press.
TRANSITION Such an extension would be hard for May to sell at home, where her overall strategy has been criticised by all sides - Brexit campaigners accuse her of making Britain a vassal state, European Union supporters say the offer is the worst of all worlds and others are increasingly frustrated over the talks.
Tusk also said the leaders expressed their resolve to stop the flow of illegal migration.
UK Prime Minister Theresa May.
London and Brussels have acknowledged that there should be a sort of safety net in place if the agreement on EU-UK partnership does not ensure that this border remains frictionless. "Such a regime should help to protect our citizens, companies and institutions from all kinds of cyber security threats", he said.
May said a proposed 21-month transition period for the United Kingdom after Brexit could be extended by "a matter of months".
Irish Prime Minister Leo Varadkar summed it up by saying "big gaps" remained between the two sides "both in terms of the shape of the future relationship and also the protocol on Northern Ireland and Ireland and the backstop".
Juncker and May, however, remain hopeful that the Brexit deadlock could be broken and an agreement reached.
"A further idea that has emerged is to create an option that extends the implementation period for a matter of months", May told reporters in Brussels on Thursday.
The group called on Mrs May to "reset" the negotiations and ditch her Chequers Brexit blueprint in favour of a Canada-style free trade agreement.
Such a border would violate the Good Friday Agreement (GFA) or Belfast Agreement that Northern Ireland's present devolved system of government is based on.
Asked if the whole deal could founder over the border issue, he replied: "My answer is yes".
European Union leaders were converging on Brussels Wednesday for what had been billed as a "moment of truth" Brexit summit but which now holds little promise for a breakthrough.
Britain's Conservative former prime minister John Major (L) stands alongside Labour former prime minister Tony Blair (R) as he speaks at the University of Ulster in Derry (Londonderry), Northern Ireland on June 9, 2016 during a visit to campaign for a "remain" vote in the European Union referendum.
Alternatively, May has suggested that the entire United Kingdom remain in the customs union of the EU for a temporary period in the hope that the "Irish problem" can be resolved later.
Prime Minister Theresa May insisted any extension to the transition period would only last "a matter of months" after a furious Tory backlash.
The French Government yesterday published draft no-deal legislation that could require British citizens to obtain a visa before travelling to France following the UK's withdrawal from the EU.
Sterling gained on the report that Britain would allow the Northern Ireland backstop, or fallback arrangement in case of failure to agree a broader deal, to have no fixed time limit.
With divorce talks stuck, the bloc has suggested extending that period, to give more time to strike a trade deal that ensures a frictionless border between Northern Ireland and Ireland.
"Today, we do not know what they want".
At present the two sides are proposing that Britain remains inside the European Union single market and is still bound by its rules from the time it leaves the bloc in March until December 2020, to give time for new trade relations to be set up.