Mrs May will make her appeal on the eve of a crunch meeting of leaders of the remaining 27 member states on Friday, when they are expected to approve a draft deal on Britain's transition to Brexit, opening the door for talks on trade.
European Council president Donald Tusk raised doubts earlier this week over whether a deal reached by Brexit Secretary David Davis and chief EU negotiator Michel Barnier would be acceptable to all 27 - with Spain reported to be holding out to improve its hand over Gibraltar.
But a senior UK official said on Thursday evening that "all the indications are positive" that the transition deal will get the thumbs-up.
Mrs May will return to the UK after dinner on the first day of the European Council summit on Thursday, leaving the other 27 leaders to discuss Brexit in her absence.
Addressing them over dinner she is expected to say that "compromise on both sides" had been required to reach a mutually acceptable legal text on the transition, which will see the UK continue to observe EU rules until the end of 2020 while winning the freedom to negotiate and ratify trade deals with outside countries.
Britain believes that moving on to trade talks with the EU will help unblock the logjam over the Irish border and prevent the need to implement a "backstop" solution proposed by Brussels which would keep Northern Ireland in the Customs Union.
London is hopeful that a solution to keep Northern Ireland's border with the Republic open will emerge in the context of a deal allowing trade between Britain and the rest of the EU to be as frictionless as possible.
Mrs May will tell the EU27 leaders: "We have the chance now to create a new dynamic in the talks, to work together to explore workable solutions on Northern Ireland, on our future security co-operation and in order to ensure the future prosperity of all our people.
"This is an opportunity it is our duty to take and to enter into with energy and ambition."
Irish Taioseach Leo Varadkar played down suggestions that he wanted the border issue settled by the summer.
"Would I like to have it done by June? Yes, absolutely," he said. "But I would rather have the right deal in October rather than any deal in June."
Mr Varadkar said he envisaged a trading relationship between the UK and the EU, "so close that many of the measures in the backstop may become unnecessary".
European Council President Donald Tusk has recommended EU27 endorsement of the transition deal, which he said would put off "the negative consequences of Brexit" for another 21 months after the formal date for the UK's withdrawal in March 2019.
Under the terms of the joint legal text agreed by Mr Davis and Mr Barnier, trade deals negotiated by the UK with outside countries during the transition period will be allowed to enter into effect on January 1 2021.
Mrs May told the House of Commons on Wednesday that for the first time in 40 years, Britain would be able to "forge our own way by negotiating our own trade agreements".
However, the PM faces warnings that the deal could be scuppered by her own MPs unless she tears up "unacceptable" proposals for fishing.
Some 14 of the PM's backbench parliamentary allies - 13 Conservatives and one DUP MP - have signed a joint letter denouncing the draft deal agreed by the Government earlier this week.