A total of 23%of CCGs and 47% of NHS trusts said they expected to end 2016/17 in deficit, the think tank's latest quarterly monitoring report found.
The proportion of CCGs predicting a deficit has doubled since this time last year.
Meanwhile, key NHS performance targets are now being missed not just in winter but all year round as demand facing the health service hits unprecedented levels, the report's authors warned.
The number of patients attending A&E departments in the first quarter of 2016/17 hit 6m, with 1m of these admitted to hospital.
NHS waiting times
Nearly one in 10 patients spent more than four hours in A&E, the highest proportion since 2003/4. In June 2016, 6,100 patients were medically fit but awaiting discharge from hospital - the highest number since data began being collected - and 3.8m patients were waiting for elective treatment, the highest number since December 2007.
King's Fund policy director Richard Murray said: 'Hospitals are treating more patients than ever before. Winter usually brings a dip in NHS performance but what is striking now is that key targets are being missed all year round. This reflects the impossible task of continuing to meet rising demand for services and maintain standards of care within current funding constraints.
'While new investment and actions taken to tackle overspending have reduced deficits among NHS providers in the first quarter of the year, it would be a mistake to suggest that the financial pressures which have engulfed the NHS have eased. Unless more is done to tackle rising demand, the ideas emerging from sustainability and transformation plans about cutting beds and reconfiguring hospitals will look even more unrealistic.'
BMA chairman Dr Mark Porter said: 'This is yet another report which shows the health service simply cannot keep up with unrelenting demand.
'Up and down the country, hospitals are being forced to close their doors, or limit access to vital services like paediatrics and emergency care due to chronic understaffing. Yet at the same time the government is trying to force through a junior doctor contract that risks fuelling the workforce crisis.
'This is crippling at a time when we need more of these doctors to keep up with rising demand on services. It also creates a vicious circle, adding to pressures on existing doctors, further increasing the risk of burnout and making these areas of medicine less attractive to doctors in training.
'The government appears to have no real solution to the funding crisis engulfing the NHS. The NHS needs a long-term plan and enough funding to deliver care for everybody in the country.'
A DH spokesman said: 'Since 2010 the NHS has dealt with significantly rising demand from our ageing population while improving the quality of care - 5.8m people were seen in A&E in the first three months of this year and the number treated within the 4-hour target continues to rise in the thousands.
'We are committed to delivering a safer seven day NHS which is why we have invested £10bn to fund the NHS's own plan to transform services in the future.'