Mrs May has, at least until now, been praised from some quarters for her "resilience" in refusing to step down despite repeated setbacks - for example, losing the Tory majority in the House of Commons in the 2017 general election, losing multiple Brexit secretaries and other senior ministers over her negotiating strategy with the European Union and the proposed deal which it produced, losing three votes on that deal in the House of Commons, having her government found in contempt of parliament for the first time in recorded history, and losing the support of more than half her backbenchers in a parliamentary party in a parliamentary party confidence vote.
No 10 has instead set a new deadline of the Commons summer recess, likely to be the end of the third or fourth week of July, to finalise Brexit and pass legislation, suggesting that they expect MEPs are all but certain to take their seats on July 2.
May had hoped to avoid taking part in the vote, but so far talks with the opposition Labour Party aimed at finding a way forward on Brexit have not succeeded in breaking the deadlock.
But the left-of-center Labour Party insists it will only agree to a Brexit deal that includes a permanent customs union with the EU to avoid barriers to the trade of goods. But those talks infuriate some Conservative backbenchers and the person they want out of a job by the summer is Theresa May.
The report cited unnamed government sources, who said the referendum plan would only become relevant if talks with Labour failed and a majority in parliament supported holding another public vote.
Earlier, May told her cabinet that last week's local elections, when the Conservatives lost hundreds of council seats, underlined the need to get on with Brexit.
Some members - even some who want May to resign - are nervous that it could look bad to alter party procedures exclusively with a view to ousting the prime minister.
Meanwhile, Prime Minister will meet with 1922 committee leader Sir Graham Brady on Tuesday.
But with both major parties, like much of the country, deeply divided over Brexit, any breakthrough from the current round of talks might be hard to come by.
Jeremy Corbyn will pay a visit to the county this week to launch the Labour manifesto for the European elections.
Foreign Secretary Jeremy Hunt said "compromises on all sides" were needed "because the message of last week was that voters for both main parties are very, very angry about the fact that Brexit hasn't been delivered".
"I want to look at whatever deal is come to between the parties and I know this is a crucial week", he told BBC radio.