If British MPs reject the plan, they might demand the government try to negotiate a new deal, but that would require more time.
Mr Mundell told the BBC: "I haven't threatened to resign".
The EU's insistence on a fail-safe for the issue of the Northern Irish border appeared to leave British Prime Minister Theresa May without negotiating options. May spoke of "working intensively over the next days and weeks" to achieve agreement that avoids a no-deal departure from the bloc for Britain on March 29 that could create chaos at the borders and in the economy.
Nobody really wants Britain to crash out of the European Union without a withdrawal deal, but time is running out and the sides have bogged down over the contentious issue of how to manage the Irish border. However, it would come at both a political and financial cost of up to £10 billion in additional European Union fees.
The possibility of an extension has infuriated hard-Brexit backers in Ms.
But May was under attack from across Britain's political spectrum after saying she was considering the European Union proposal for a longer post-Brexit transition period.
But Lithuanian president Dalia Grybauskaite said there would be "no breakthrough" this week because May lacked a strong mandate from her party and parliament.
After amendment, this agreement was supported by the DUP and by David Davis and Boris Johnson. I've done my bit.
"She is losing the confidence of colleagues of all shades of opinion".
The backstop is a proposed mechanism to keep the United Kingdom in the EU's Customs Union after Brexit - essentially precluding the country from taking back control of its global trade policy - in order to prevent a hard economic border between the British province of Northern Ireland and EU member the Republic of Ireland. One leading Conservative Remainer told BI: "It's daft".
May committed past year to keeping the border open between the two parts of the island of Ireland, even if Britain leaves the bloc without an exit agreement or a free trade deal.
Arriving in Brussels, Mrs May insisted there had been "very good progress" since last month's meeting in Salzburg, when she was humiliatingly told that her Chequers plan for Brexit would not work.
His comments follow Theresa May's suggestion that the idea of extending the Brexit transition period "for a matter of months" had been considered. "We need time, we need much more time..."
But the idea is controversial among May's eurosceptic Conservative MPs, who fear it is simply a way of delaying Britain's exit. Brexiteers are making all the noise for now, but Remainers won't accept a deal they believe to be worse than the status quo. "If not, what's the point?" they said.
The EU and the United Kingdom have agreed they do not want a return to a hard border, where passport and customs checks would be returned, but they can not agree how to achieve that.
"Brexit offers the prize of a better future, global free trade deals and political independence", said the letter. "If we were to go in that direction, it would come with a certain number of conditions".
Analysis: May has little choice but to extend.
Extending the transition period would mean that the United Kingdom would prolong the period where it has no say in EU policymaking but would still have to pay into the European Union budget.
Meanwhile, the top European Union trade official played down the importance of a USA announcement that it will pursue a trade deal with Britain, noting London can not negotiate such pacts until it leaves the bloc. May's proposal to extend this period is therefore entirely sensible, even if the realisation that such an extension is needed has come very late in the process.
A source close to Mr Mundell said he wants "assurances any extension to the transition would not delay exit from CFP beyond the agreed date".
In a speech in London on Tuesday evening, John Major, a former Conservative Prime Minister, said the Brexit vote was a "colossal misjudgment" and that Brexit-backers sold the country a fantasy and would not be forgiven.
FILE - A vehicle crosses over the border from the Irish Republic into Northern Ireland near the town of Jonesborough, Northern Ireland, Jan. 30, 2017.
But there would have to be new regulations in place, he said, without giving further details.
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