A survey of elected grassroots Conservatives found three-quarters of the Prime Minister's councillors want her to resign, with 43 per cent calling for her to quit immediately.
The Survation study, commissioned by the Mail on Sunday, found that just over half (52 per cent) would vote Tory at the European election, which would rise to 65 per cent if Boris Johnson replaced Mrs May.
Some 40 per cent said they would vote for Mr Farage's rival Brexit Party, a figure which would nosedive to 22 per cent if Mr Johnson was Prime Minister, the poll found.
Councillors across the party were almost in complete agreement that the Tory party has been damaged by Brexit, with 96 per cent of councillors saying the party had been harmed.
One councillor in the study slammed Mrs May's leadership amid the Brexit turmoil, saying: "The Conservative Party is dead. It will take a strong leader to dredge it out of the mud."
Another added: "For God's sake get on with it (Brexit) - it is killing us on our doorstep."
Survation questioned 781 Tory councillors between April 17 and 19.
Some 43 per cent said Mrs May should resign now and 33 per cent once a Brexit deal has been reached - the timetable the Prime Minister has indicated for her departure.
Mr Johnson was backed by 23 per cent as the best leader, followed by Michael Gove on 14 per cent, Jeremy Hunt on 12 per cent, Sajid Javid on 11 per cent and Dominic Raab on 9 per cent.
The Mail On Sunday report came after a survey of more than 1,000 Tory members by the influential ConservativeHome website found nearly eight out of 10 want Mrs May to quit.
And it followed the Tory group on Derbyshire County Council announcing it would not campaign for the party in the European elections.
Writing in the Sunday Telegraph, Mr Farage said that "millions of natural Tory voters have said emphatically that they will never vote Conservative again. And my strong sense is that they mean it".
He added: "Far from being seen as the party to solve the EU question, the Tories are now actively regarded by a growing number of people as being a major part of the problem.
"Under Theresa May's stewardship, there has been near-anarchy on the Tory benches in Parliament for the past couple of years, as MPs have deliberately gone against the party's manifesto and done everything they can to thwart the referendum result.
"Her own chicanery makes it unclear whether she has merely tolerated this behaviour or encouraged it."
In a sign of the pressure on Mrs May, the Sunday Times reported that she will be told within days that she must step down by the end of June or face a fresh effort by MPs to oust her.
The newspaper reported that Sir Graham Brady, chairman of the backbench 1922 Committee, will tell her that 70 per cent of Tory MPs now want her to resign.
Mrs May survived a confidence vote by her MPs in December and the party's rules prevent another attempt to force her out within 12 months.
But discontent with Mrs May's leadership has fuelled speculation that those rules could be changed to allow a fresh challenge.
The ConservativeHome survey of 1,132 party members on its panel found 79 per cent wanted Mrs May to quit and trigger a leadership contest.
The Prime Minister has said she will not lead the Tories into the next election - due in 2022 - and has promised to bring her departure forward if her Brexit deal gets through Parliament.
Former Tory MP Paul Goodman, the website's editor, said: "It's like Groundhog Day for the Prime Minister - but in which each one is worse than that before.
"Last month, a record total of seven out of 10 of our panel members wanted a new Party leader. This month we have a new record of nearly eight out of 10."
Just 19 per cent of those surveyed wanted Mrs May to stay on as leader.
Mr Goodman said: "The second postponement of Brexit and the talks with Jeremy Corbyn are undoubtedly huge contributors to this lamentable rating.
"The latter especially is making campaigning uphill work indeed for many local government candidates.
"However, we suspect the biggest factor is the European Parliamentary elections that are due to take place."
In a suggestion that ministers should now tell Mrs May her time is up, he said: "The voluntary Party has lost confidence in the Prime Minister. Is anyone listening around the Cabinet table?"
Barry Lewis, who has led the Derbyshire authority since May 2017, said he had informed party chairman Brandon Lewis of the boycott of the European elections, which he said had received overwhelming support from his colleagues.
He said: "We were promised, following the largest public mandate a UK Government has ever received, that we would be out by March 29.
"The Prime Minister said we would be out by that date countless times, so did many others in Government, and yet here we are racing towards the end of April and facing an increased prospect of participating in a European election in May that should not be happening."
As an extension to the Brexit process has been granted until October, the UK is obliged to elect 73 MEPs to the European Parliament who will sit until Britain leaves on October 31 or when the Withdrawal Agreement is ratified in the Commons.