May, who won the top job in the turmoil that followed the 2016 referendum, has sought to negotiate a Brexit deal that ensures that the United Kingdom leaves in the smoothest way possible.
But Brexit minister Dominic Raab resigned on Thursday over her deal, sending the pound tumbling. Mutinous lawmakers in her own party openly sought to challenge her leadership and bluntly told her that the Brexit deal would not pass parliament.
May, who has vowed to stay on as prime minister, was asked by a caller on an LBC radio phone-in on Friday to "respectfully stand down". She did not immediately address that part of the caller's question.
"I haven't appointed a new Brexit Secretary yet but of course I will be doing that over the course of the next day or so," May said when asked if she had offered it to Michael Gove, the most prominent Brexit-supporting minister in her government.
Gove, 51, gave no comment when asked outside his house whether he would support May. The BBC said May had offered him the job of Brexit minister but he had rejected the job.
Media reports, citing sources close to Gove who torpedoed former foreign minister Boris Johnson's leadership bid in 2016 at the last minute, will stay as environment minister.
Sterling, which has see-sawed on Brexit news since the referendum, was flat at $1.2783 on Friday, down almost three cents since a deal was struck on Tuesday.
Brexit will pitch the world's fifth largest economy into the unknown. Many fear it will divide the West as it grapples with both the unconventional U.S. presidency of Donald Trump and growing assertiveness from Russia and China.
Amid the deepest political turmoil since the Suez canal crisis, when in 1956 Britain was forced by the United States to withdraw its troops from Egypt, the ultimate outcome remains uncertain.
Scenarios include May's deal ultimately winning approval; May losing her job; Britain leaving the bloc with no agreement; or even another referendum.
To leave the EU on the terms of her deal, May would need to get the backing of about 320 of parliament's 650 lawmakers. The deal is due to be discussed at an EU summit on November 25.
Some lawmakers in May's Conservative Party have said they have submitted letters of no confidence. When 48 letters are submitted to the party's so-called 1922 committee, she will face a leadership challenge.
Politicians, officials and diplomats in London openly questioned how long May had left as speculation swirled around London that a leadership challenge could come soon.
Sky said government whips, who enforce discipline in the party, had been summoned to parliament as a challenge was close. If a confidence vote is called among her lawmakers, May would need a simple majority of the total votes in order to win.
By seeking to preserve the closest possible ties with the EU, May has upset her party's many advocates of a clean break, and Northern Ireland's Democratic Unionist Party (DUP), which props up her minority government.
The Daily Telegraph newspaper reported that the DUP had demanded May be replaced as prime minister.
"Oh I haven't had a testy exchange with Arlene," May said. "They've raised some questions with us, they've raised some concerns with us and yes we are looking at those.
"We are still working with the DUP," she said.
Nightmare for business?
The EU and Britain need an agreement to keep trade flowing between the world's biggest trading bloc and the United Kingdom, home to the biggest international financial centre.
May said she felt the threat of a no deal Brexit personally as she was Type 1 diabetic: "I depend on insulin every day. My insulin is produced by a country elsewhere in the European Union."
British aero-engine maker Rolls-Royce said it was continuing with its contingency plans.
The plans include "buffer stocks so that we have all the logistical capacity that we need to carry on running our business," said Chief Executive Warren East.
Supporters of Brexit say that while the divorce might bring some short-term instability, in the longer term it will allow the United Kingdom to thrive and also enable deeper EU integration without such a powerful reluctant member.
Meanwhile, proponents of closer relations with the EU in her own party and the Labour opposition say the deal squanders the advantages of membership for little gain.
"It is ... mathematically impossible to get this deal through the House of Commons. The stark reality is that it was dead on arrival," said Conservative Brexit-supporting lawmaker Mark Francois.
More than two years after the United Kingdom voted to leave the EU, it is still unclear how, on what terms or even if it will leave the EU as planned on March 29, 2019.