NHS Providers said the cap, which limits pay rises to 1% a year to 2019, was causing severe recruitment and retention problems in England.
The body, which represents NHS trusts in England, said the next government must look at the issue immediately.
Labour says it would look to increase pay, but the Tories and Lib Dems have not yet set out any pay plans.
Labour wants to increase pay so it better reflects the cost of living, but has not said by how much.
Over the weekend the Lib Dems did announce they would increase income tax by a penny-in-the-pound to boost investment in the NHS.
Last year the Public Accounts Committee warned that the NHS in England was 50,000 people short of the front-line staff it needed - about 6% of the workforce.
NHS Providers chief executive Chris Hopson said his members were now worried the situation was so bad that services were at risk.
"Growing problems of recruitment and retention are making it harder for trusts to ensure patient safety," he said. "Unsustainable staffing gaps are quickly opening up."
He said the seven years of pay restraint, combined with stressful working conditions, had taken a toll on the workforce.
"Pay is becoming uncompetitive," added Mr Hopson. "Significant numbers of trusts say lower paid staff are leaving to stack shelves in supermarkets rather than carry on working in the NHS."
He added that uncertainty surrounding Brexit meant that "vital recruitment from EU countries is dropping rapidly", saying: "Pay restraint must end."
While NHS Providers represents only English hospital, ambulance, and mental health trusts, any move on the pay cap would have implications elsewhere as pay recommendations are made on a UK-wide basis.
The Royal College of Nursing (RCN) is polling its members on whether they should strike over the pay cap.
It says a combination of pay freezes and caps on pay rises since 2010 have, in effect, led to a 14% pay cut due to the rising cost of living.
RCN general secretary Janet Davies said: "The government cannot ignore this warning from hospital bosses - poor pay for NHS staff damages patient care.
"If it now pays more to stack supermarket shelves than work on the wards, ministers should hang their heads in shame."
Lib Dem health spokesman Norman Lamb refused to be drawn on whether his party would increase pay.
But he accused the Tories of "running the health service into the ground".
A Conservative party spokesman said: "The truth is that in order to continue to invest in the NHS, grow staff numbers and pay and improve patient care, we need to secure the economic progress we've made and get a good Brexit deal.
"That is only on offer at this election with the strong and stable leadership of Theresa May."
Every 1% increase in pay would cost about £500m.