Wales has already adopted an opt-out approach to organ donation - in which patients are assumed to be willing to donate their organs unless they register their objection - but England, Northern Ireland and Scotland have maintained an opt-in approach.
BMA leaders have long advocated a switch to presumed consent for organ donation, warning that the current model leaves much of the UK facing an unnecessary shortage of organs that could save lives.
The poll found that 65% of 2,011 people surveyed would back an opt-out system. It also found that 66% of people were prepared to donate their organs at death, but that just 39% were signed up to the organ donation register.
The system proposed by the BMA would still allow families to stop a relative's organs being used if they believed there was an unregistered objection on their part.
BMA ethics committee chair and former GPC chair Dr John Chisholm said: 'Although organ transplantation has seen amazing medical achievements it has not yet reached its full life-saving and life-transforming potential.
'These figures show that in the current system, a large number of people who wish to donate their organs are not signing up to the register. Vital opportunities to save people’s lives are being missed.
'Around 10,000 people in the UK are in need of an organ transplant, with 1,000 people dying each year while still on the waiting list. As a doctor it is difficult to see your patients dying and suffering when their lives could be saved or dramatically improved by a transplant.
'Since soft opt-out was adopted in Wales, 160 organs have been transplanted, almost a quarter of which were down to the new system.
'The BMA is calling for all UK governments to follow suit and adopt a soft opt-out system. If we have an opportunity to address the chronic shortage of organs and save the lives of patients across the UK, surely we should be taking it.'