By Elizabeth Piper and William Schomberg
LONDON (Reuters) - British Prime Minister Theresa May is facing a new push by members of her Conservative Party to lead Britain out of the European Union in the next few months, even if it means a potentially damaging no-deal Brexit.
Conservative lawmakers urged May against a long extension of the Brexit process in a letter sent after her exit deal was rejected for a third time by the House of Commons on Friday, a lawmaker said.
The Sun newspaper said the letter was signed by 170 of the 314 Conservative lawmakers in parliament, including 10 cabinet ministers, heaping more pressure on May, who has offered to resign if she can get her Brexit deal through parliament.
Friday's vote, on the day that the country was originally due to leave the bloc, kept Britain in political turmoil.
May has less than two weeks to convince the 27 other EU countries that she can solve the impasse. Otherwise she will have to ask the bloc for a long extension or take Britain out of the EU on April 12 with no deal to soften the economic shock.
"We want to leave the EU on April 12 or very soon afterwards," The Sun quoted a Brexit-supporting minister as saying in reference to Friday's letter.
A spokeswoman for May declined to comment on the letter.
The Conservative Party's chairman Brandon Lewis said he was aware of it but had not seen it.
Many eurosceptic Conservative lawmakers have long demanded what they say would be a clean break from the EU.
By contrast, the 650-seat lower house of parliament has voted overwhelmingly against a no-deal Brexit although its view is non-binding.
On Monday, lawmakers will try to agree on an alternative to May's Brexit plan. The options that have so far gathered most support involve closer ties to the EU and a second referendum.
The BBC said government officials had not ruled out the possibility of a run-off vote in parliament next week between the most popular option proposed by lawmakers and May's deal.
Lewis, the Conservative Party chairman, said all options were on the table but that a customs union with the EU -- an idea backed by the opposition Labour Party and some Conservatives -- would be difficult.
"We've got to look at what we can do next and we have to do something different," he told BBC radio when asked if May might try to put her withdrawal agreement to a parliamentary vote for a fourth time.
"Parliament will continue this process on Monday and we've got to look at all the options."
A customs union would fly in the face of the Conservative Party's pledges before the 2017 national election and would not respect the outcome of the 2016 Brexit referendum, Lewis said.
When asked if a way out of the impasse was to hold a fresh national election, he said he did not think British voters wanted to return to the polls.
(Reporting by Elizabeth Piper, William Schomberg and Elisabeth O'Leary; Editing by Catherine Evans)