Video report by ITV News Political Correspondent Carl Dinnen
Doctors have rejected a demand to open GP surgeries seven days a week and accused Theresa May of using them as "scapegoats."
The Government has urged GPs to roll out a seven-day service to help relieve pressure on crisis-hit A&E departments.
Downing Street also threatened to withdraw funding if surgeries didn't move to 8am to 8pm opening times - unless they could prove there is no demand from patients.
But the British Medical Association (BMA) has hit back angrily and accused ministers of trying to "scapegoat" doctors for the crisis in the NHS.
The BMA argued the request was simply masking chronic under-funding in the healthcare system.
The government has said the NHS is already committed to bringing in 8am to 8pm services in general practice seven days a week, backed by an extra £528 million per year in funding by 2020/21.
But GPs have called it "unrealistic" and warned doctors are already "spread too thin" and quality of care will suffer.
Dr Chaand Nagpaul, the BMA GP committee chairman, said they were facing a "major alert" with one in three GP practices reporting unfilled vacancies.
Eight in 10 claimed they were unable to provide safe care.
"We have got a very serious problem that we don't have the capacity in general practice," Dr Nagpaul told the BBC Radio 4 Today programme.
"The crisis in the NHS won't be solved by scapegoating or deflecting blame on to GPs.
"All patients can see a GP 24 hours a day, 365 days a year. Patients can see a GP when they need to.
"What we cannot be providing is stretching a service where we are struggling - and we do manage in spite of the pressures - into a seven-day service."
Professor Helen Stokes-Lampard, the chairwoman of the Royal College of GPs, said the Prime Minister's intervention was "extremely unfortunate" and the plans were "misguided".
"It is not the case that GP surgery routine opening hours are contributing to the pressures our colleagues in A&E departments are currently facing," she said.
She added: "It has never made sense to force GPs to offer services that there is little patient demand for.
"In many cases practices have already had to actually stop offering extended opening hours because of a lack of patient demand for them."
The row comes amid frustration among ministers that the unavailability of GP appointments is driving patients to seek treatment in hard-pressed hospital accident and emergency departments.
The director of acute care for NHS England, Professor Keith Willett, has recently estimated that 30% of the patients attending A&E would be better cared for elsewhere in the system.
Meanwhile the latest official figures showed more than four in 10 hospitals in England declared a major alert in the first week of the year as services came under increasing pressure.
A Downing Street source said: "Most GPs do a fantastic job, and have their patients' interests firmly at heart.
"However, it is increasingly clear that a large number of surgeries are not providing access that patients need - and that patients are suffering as a result because they are then forced to go to A&E to seek care.
"It's also bad for hospitals, who then face additional pressure on their services."
The proposed reforms include:
GPs to meet the extended hours commitment of 8am to 8pm seven-days a week unless they can prove there isn’t the demand
Extended hours should be offered when patients want them including evening and weekend appointments
GPs accessing extended hours funding should expand their online services to free up consultation and treatment time in surgeries
Extra funding will be contingent on demonstrating that they offer and advertising flexible appointments
Practices could be asked to use a new appointments tool to submit data on the number of appointments offered to understand demand