Writing in the UK’s Sunday Express newspaper, May urged MPs not to let down the Britons who voted in favour of exiting the European Union in a June 2016 referendum.
“Doing so would be a catastrophic and unforgivable breach of trust in our democracy,” she wrote.
“So my message to Parliament this weekend is simple: it is time to forget the games and do what is right for our country.”
The UK is poised to leave the EU on March 29, two years after it triggered Article 50, the exit clause in the EU’s constitution, and kick-started arduous negotiations with European leaders over a divorce deal.
But May is facing an uphill task to save her withdrawal agreement – forged during 18 months of talks between London and Brussels – from a crushing defeat in parliament. MPs are due to vote on May’s deal on Tuesday.
The deal has come under fire from across the political spectrum, with opponents of the EU seeking a cleaner break and pro-European legislators pressing for a second vote on membership of the bloc.
The main opposition Labour Party, which is in favour of remaining in a permanent customs union with the EU, has suggested it will seek a no-confidence vote in the government if MPs throw out May’s withdrawal plan in a bid to trigger a general election.
“We will table a motion of no confidence in the government at a time of our choosing, but it’s going to be soon, don’t worry about it,” Labour Party leader Jeremy Corbyn told the BBC on Sunday.
If the government loses a no-confidence motion, there will be a period of 14 days in which parties can seek to find an alternative working majority in parliament.
If they fail to do so, a general election would be called.
Corbyn said that if he forced an election and his party won, Brexit may have to be delayed while a new government negotiated a revised deal with the EU.
“An election would take place what February-March time, clearly there is only a few weeks then between that and the leave date, there would have to be a time for those negotiations,” he added.
EU leaders have repeatedly insisted that the withdrawal agreement cannot be renegotiated.
May pulled a vote on the deal last month, acknowledging it would have been roundly rejected by MPs.
She has since lobbied European counterparts over the deal’s contentious “Irish backstop” clause in a bid to win over parliamentary critics.
The safety net “backstop” measure would guarantee no hard border is reintroduced on the island of Ireland in the event that post-Brexit trade negotiations between the UK and the bloc prove unsuccessful.
Those opposed to the clause argue it could tie the UK into the EU’s orbit indefinitely.
In the event May’s deal is rejected on Tuesday, her government will have just three days to bring a revised plan back to Parliament.
Following a vote in the Commons last week, MPs will also have the right to amend any second proposal brought forward by the British leader.
That could open the way for several different outcomes, ranging from a so-called “managed no-deal” exit to another referendum.
Both Corbyn and May are opposed to a second popular vote, despite recent opinion polling suggesting a majority of Britons believe the final decision about Brexit should be made by a new referendum.