The remarks follow a big Government defeat in the House of Lords on the issue of retaining a customs union with the EU last week and reports the Prime Minister and her inner team are considering a rethink on the matter.
As well as this 62% said Britain should continue to be part of the single European market.
The confirmation comes ahead of a symbolic parliamentary debate on the issue later this week.
Rebel Tory MPs are preparing to back a cross-party motion later this week demanding the Government negotiates a customs union with Brussels. The president of the Confederation of British Industry's (CBI) meanwhile, told Javid off for neglecting evidence that leaving the customs union would damage British businesses.
Labour has called for the United Kingdom to join a new customs union after Brexit, saying it would leave the current one but negotiate a treaty afterwards that would "do the work of the customs union".
Foreign Secretary Boris Johnson, Brexit Secretary David Davis and Trade Secretary Liam Fox are said to be pushing to leave the customs union and arrange a union with the EU from outside.
Earlier this year, a leaked government report predicted that leaving the customs union would reduce the growth of the United Kingdom economy by up to 8% over the next 15 years.
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Senior Brexit-supporting ministers are likely to make their case for breaking free of the European Union rules directly to the prime minister when she holds a key meeting of her inner cabinet on Wednesday, one official said.
May is set to face calls from leading Brexiteers Boris Johnson, David Davis and Liam Fox to abandon her preferred form of a customs deal with the European Union, according to The Times.
Common external tariffs are applied to goods imported from countries from outside the bloc.
Among those who said they voted Leave, 43% are unhappy with the status of negotiations compared to 38% who were happy.
The Confederation of British Industry has called for the United Kingdom to remain in the customs union, while the Labour Party has also backed a similar arrangement.
The government says the rights the charter protects are already covered by British law and May's spokesman said earlier on Monday the withdrawal bill, which was approved by the House of Commons before being sent to the upper house, was the best way of providing "the smoothest possible Brexit".
On Thursday, Conservative supporters of remaining in the Customs' Union, including Anna Soubry and Kenneth Clarke, will argue their case. The second, known among civil servants as the "maximum facilitation" option, would seek to minimise but not totally eliminate checks on the Irish border.