The YouGov survey found just 39 per cent of people backed a hard Brexit, even if it meant having no free trade agreement with the EU, while 47 per cent backed a softer approach.
It comes as the Prime Minister is due to give a major speech on her strategy for withdrawal negotiations, with reports suggesting she will say she is prepared to leave the single market and customs union.
The survey showed 15 per cent of the public preferred maintaining "the best possible trade links" with EU, even if it meant adhering to some rules and giving up some control over immigration, while 10 per cent backed a form of associate membership and 23 per cent backed remaining a full member.
The total, 47 per cent, who preferred one of the more emollient options, outnumbered the 39 per cent who said they wanted full control of borders regardless of whether that meant no free trade deal with the European Union.
Reports have suggested Ms May will indicate in her speech tomorrow that the Government is prepared to abandon the single market, customs union and the European Court of Justice in order to achieve a clean break with the EU.
In a blow to Jeremy Corbyn, the YouGov poll also showed that 42 per cent of people are unsure about what the Labour party's priorities for Brexit are.Reuse content