Mrs May has appealed to the British public to get behind the agreement - saying that although it involved compromises, it was a "good deal that unlocks a bright future for the UK".
Harry Shindler, who lives in Italy, and 12 other British expats questioned the legality of the Brexit referendum in 2016 because they and more than a million Britons living in the European Union did not get to have their say due to the 15-year rule on voting rights for expats.
Lidington gave a hint of the arguments he is likely to use in an intriguing tweet on Sunday, describing the deal as a "good compromise" between leave and remain.
Foreign Secretary Jeremy Hunt said Britain was getting "between 70 percent and 80 percent" of what it wanted but warned parliamentary approval was "looking challenging".
May, when asked by reporters, if she would consider stepping down if the deal fails in Parliament, May said "this is not about me".
Leaders of the European Union approved a Brexit deal with the United Kingdom on Sunday, clearing - at least on Europe's end - the path for the U.K.to formally exit the bloc in the end of March.
Spain will revive its bid for shared sovereignty over Gibraltar once Britain has left the European Union, the country's prime minister said on Sunday, adding Madrid had the support of the bloc to resolve the 300-year-old dispute.
"I don't believe that, so far, this deal delivers on what the British people really voted for", he told Sky's Sophy Ridge show.
Britain will leave the European Union on March 29 but remain inside the bloc's single market and be bound by its rules until the end of December 2020, while the two sides work out a new trade relationship.
British Foreign Secretary Jeremy Hunt said Sunday that the United Kingdom faced "very big risks" if the deal was rejected, according to the Reuters news agency.
Former Conservative leader Iain Duncan Smith said he would find it "very, very difficult" to support the agreement. "They failed in the Withdrawal Agreement, and they failed again in the Political Declaration", May said.
As for what the deal entails, it does allow for an orderly Brexit, but in doing so just preserves the status quo for nearly two more years.
May insists her deal delivers on the things that matter most to pro-Brexit voters - control of budgets, immigration policy and laws - while retaining close ties to the U.K.'s European neighbors. "Those who think that by rejecting the deal that they would have a better deal will be disappointed the first seconds after the rejection".
One senior Conservative told the Sunday Times that any attempt to force a hard no-deal Brexit would lead to a mass walkout from Cabinet: "If she said she'd go for mitigated no deal, she would lose most of her cabinet".
"In Parliament and beyond it I'll make the case for it with all my heart and I look forward to that campaign", she declared, but also twice refused to say whether she would resign as prime minister if she is not successful in the effort.
He said British lawmakers would see that "the alternative is a no deal, cliff edge Brexit which is something of course that we all want to avoid".
There is little evidence Britain's embattled prime minister will have sufficient support to win legislative endorsement of the deal in a House of Commons vote next month.