The survey of 522 family doctors for GP magazine Pulse found 16% had been asked to refer patients.
It comes after Pulse reported concerns among GP leaders that practices are being put in an "impossible position" by charities that require a referral before they will offer help.
Such referrals are aimed at making sure support reaches the most needy and can come from places such as schools, GPs and job centres.
Senior GPs have said the system can put a strain on the doctor-patient relationship, as well as taking up appointments.
There has also been a 21% reported increase in requests for GPs to verify work capability due to cuts to the welfare system, according to figures from last year.
Professor Clare Gerada, former chair of the Royal College of GPs, said family doctors were being caught up in the "hoops" the genuinely needy had to jump through to get help.
She said: "Poverty is an enormous workload issue and, again, it's the inverse care rule because it creates more work for GPs in poorer areas who don't get resourced for it - so you end up with more work and less time.
"We're there trying to sort out everybody's problems and meanwhile the posh middle classes are complaining because they can't get access to us."
Chris Mould, chairman of the Trussell Trust network of food banks, said: "GPs should have the ability to refer to a food bank when they come across a patient who they believe needs a food bank for health reasons, especially as levels of malnutrition are reported to be increasing.
"GPs should not, however, be placed in a position to assess whether someone needs a food bank when the crisis is not health-related and they do not have enough information to make an accurate assessment of a patient's situation."
Pulse editor Steve Nowottny said: "That a significant number of patients are now going to their GP asking to be referred to a food bank is clearly a concern."