Theresa May has asked MPs to make an "honourable compromise" as she seeks to persuade them to back her Brexit deal at the third time of asking.
Writing in the Sunday Telegraph, the prime minister said failure to support the deal would mean "we will not leave the EU for many months, if ever".
Mrs May is expected to bring her withdrawal agreement back to the Commons next week for a third vote.
The EU will decide the terms and conditions of any extension. Legally, the UK is still due to leave the EU on 29 March.
Meanwhile, Labour leader Jeremy Corbyn has written to MPs across the Commons inviting them for talks to find a cross-party compromise.
Mrs May says if Parliament votes for her withdrawal deal before an EU leaders' summit on Thursday, the UK will seek a short delay to Brexit to pass the necessary legislation.
"That is not an ideal outcome - we could and should have been leaving the EU on March 29," she said.
"But it is something the British people would accept if it led swiftly to delivering Brexit. The alternative if Parliament cannot agree the deal by that time is much worse."
If a deal is not agreed before Thursday, EU leaders are contemplating a much longer delay.
Mrs May said it would be a "potent symbol of Parliament's collective political failure" if a delay to Brexit meant the UK was forced to take part in May's European elections - almost three years after voting to leave the EU.
On Tuesday, MPs overwhelmingly rejected Mrs May's withdrawal agreement for a second time - this time by 149 votes.
In her article, Mrs May said she has more to do to convince dozens of Tory MPs to back the deal - as well as getting the Democratic Unionist Party to drop their opposition.
She wrote: "I am convinced that the time to define ourselves by how we voted in 2016 must now end.
"We can only put those old labels aside if we stand together as democrats and patriots, pragmatically making the honourable compromises necessary to heal division and move forward."
The DUP, which has twice voted against the agreement, said there were "still issues to be discussed" and it remained in talks with the government.
The 10 votes provided by the DUP, which props up the government, are thought to be key to the prime minister securing her deal.
Former Cabinet minister Esther McVey, who resigned over the Brexit agreement, has suggested fellow Brexiteers could back Mrs May's "rubbish" deal next week to make sure the UK leaves the EU.
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Mr Corbyn has offered talks with opposition leaders and backbench MPs in an effort to find a Brexit compromise which could replace Mrs May's plan.
The Labour leader has invited Liberal Democrat leader Sir Vince Cable, DUP deputy leader Nigel Dodds, SNP Westminster leader Ian Blackford, Plaid's Liz Saville Roberts and Green MP Caroline Lucas.
In his letter, he called for urgent meetings to find a "solution that ends the needless uncertainty and worry" caused by Mrs May's "failed" Brexit negotiations.