Prime Minister David Cameron is struggling to convince voters he is telling the truth about why Britain should stay in the European Union and his main "Out" rival Boris Johnson is doing a better job, an opinion poll found.

Only 21 percent of respondents in the survey carried out by polling firm ComRes agreed that Cameron was more likely to tell the truth about the EU than Johnson while 45 percent said Johnson was more believable than Cameron.

With less than six weeks to go until the June 23 referendum on Britain's EU membership and voters evenly split on how they intend to cast their ballots, the rival camps have stepped up campaigning.

Cameron has warned of the risk of a hit to Britain's economy from a decision to leave the world's biggest trade bloc. Johnson says Britain would flourish outside the EU if allowed to make its own rules, strike its own trade deals and spend its EU budget contributions at home.

The ComRes poll, conducted for the Sunday Mirror newspaper and the Independent website, found 33 percent of respondents believed they would be better off if Britain stayed in the EU, only slightly more than the 29 percent who thought they would be better off if Britain left.

ComRes interviewed 2,043 adults online on Wednesday and Thursday.

Bank of England Governor Mark Carney warned on Thursday of the risk of a sharp slowdown in Britain's economy, and possibly a short recession, if the country left the EU.

The ComRes poll did not ask voters how they intended to vote on June 23.