The key challenge, which she will discuss with European Commission President Jean-Claude Juncker in Brussels on Thursday, stems from the future status of the border between the Republic of Ireland (which will remain in the EU) and Northern Ireland (which is scheduled to leave the Brussels-based club along with England, Scotland and Wales at the end of next month).
Assuring Irish Prime Minister Leo Varadkar of the solidarity of all other member states as Dublin insists Britain give legal guarantees to avoid disrupting the Northern Ireland border, Tusk told a joint news conference in Brussels that he saw no force capable of blocking a "pro-Brexit" government and opposition.
Speaking on Tuesday the prime minister's spokesman said: "Following the vote, work has been taking place on all of the options and it is important for that to take place before we go to Brussels".
Tusk and Varadkar underlined that preparations are being intensified for a "no-deal" scenario under which Britain would leave without an agreement; a possibly disastrous development that could inflict heavy economic and political damage in the United Kingdom and the European Union alike.
Plans for Britain to leave the European Union on 29 March under a withdrawal agreement signed a year ago were thrown into doubt when British lawmakers rejected the accord.
May is now discussing "alternative arrangements" to the backstop with members of her own party.
At this critical juncture, Northern Ireland is represented only by the Democratic Unionist party, which is not representative of the people on Brexit - and so much more.
EU Brexit negotiator Michel Barnier said that backstop remains "the only operational solution available" for an orderly exit of Britain from the EU.
Mrs May will seek to secure changes which can persuade MPs to support her deal in a series of votes expected on February 14.
"From a political point of view, there is still time", Merkel said.
As the party that negotiated the Good Friday agreement, which brought peace to Northern Ireland, the Labour Party has a solemn duty to be responsible in our approach to the effects of Brexit on Northern Ireland and the Good Friday agreement - no matter how great the desire to see a change in government, we can not forget our responsibility to the peace process and to protecting its integrity.
In any of the above cases, the extension of article 50 is highly probable, but the probability of a second referendum or revocation of the Brexit is reduced.
Critics of the backstop argue its lack of any agreed time-limit is unacceptable as it could see the United Kingdom locked into a customs union deal with the EU indefinitely and Northern Ireland kept under EU single market rules.
The lack of a clear road map led to a change to the Prime Minister's initial position ruling out renegotiation of the agreement with Europe and led her to support the amendment of the Conservative MP Brady in this regard, which were approved together with the help of the Unionists from Northern Ireland.
The EU, which has long regarded the border as the thorniest issue in Brexit talks, is also adamant that the backstop can't be removed.