Arlene Foster has indicated that Northern Ireland staying part of the UK takes priority over leaving the EU customs union.
The DUP leader has made it clear she believes that the UK should leave the customs union.
But she said the DUP's "red line" is that Northern Ireland should follow the rest of the UK, whatever happens.
With five months to go until a Brexit deal is supposed to be set, the UK still has no clear position on exactly what customs arrangements it wants with the EU after withdrawal.
Mrs May is said to favour a "customs partnership" which favours closer customs ties with the EU, to avoid a hard border in Ireland, or a "streamlined arrangement" with loser customs ties, but a harder border.
Eurosceptic Cabinet ministers thought they had "killed off" the customs partnership plans after a meeting of the Brexit war cabinet last week, amid concerns it would see a climbdown over Brexit and Britain staying in the customs union.
However, Mrs May has insisted the partnership is still on the table, with the issue set to be decided at another meeting of the Brexit war cabinet on Thursday.
Mrs Foster was asked on yesterday's The Andrew Marr Show on BBC One if she had discussed issues surrounding Brexit - such as staying in the customs union to prevent a hard border in Ireland - with Mrs May.
"We have continuous conversations around this issue and I spoke to the Prime Minister yesterday afternoon," the Fermanagh MLA said.
"We want to see a sensible Brexit, by respecting the vote of the British people and leaving the customs union.
"We don't want what the EU have put on the table, a red line down the Irish sea; what we seek to do is to protect the integrity of the United Kingdom from a constitutional and economic point of view."
Mr Marr asked if staying in the customs union would be a price worth paying for a soft border. "The British people voted to leave the customs union, we don't believe we have to stay in it to have free flow between ourselves and the Republic of Ireland," Mrs Foster replied.
"Proposals have been put forward by the government and they have been dismissed out of hand by the EU.
"We would like to see less rhetoric and more engagement.
"As Nigel Dodds said, our fundamental red line is we have to move in step with the rest of the United Kingdom."
Last week, Mr Dodds said the DUP would not accept the EU's 'backstop' option - which would effectively draw a customs border down the Irish Sea by keeping Northern Ireland in the customs union. He told PoliticsHome: "For us that would be a red line, which we would vote against the government, because you might as well have a Corbyn government pursuing openly its anti-unionist policies as have a Conservative government doing it by a different means."
Some have interpreted this as the DUP throwing Mrs May a lifeline, by not ruling out the whole of the UK remaining in some form of customs union - as long as Northern Ireland is not treated differently.
The DUP props up Mrs May's government in Westminster in a confidence and supply agreement.
Mr Marr suggested to Mrs Foster that the DUP would not bring down the government because they would not do anything to help Jeremy Corbyn become Prime Minister.
"We do think Jeremy Corbyn is beyond the pale due to his support for the IRA in the past, and as someone who's suffered as a result of IRA terrorism it is a very personal issue for me," Mrs Foster said.
"In terms of bringing down the government, we don't need to bring down the government, because Theresa May was very clear about her unionist credentials long before she got involved with the DUP."
Yesterday, Sinn Fein MEP Martina Anderson said the DUP is continuing to ignore the views of people In Northern Ireland on Brexit. "Once again Arlene Foster has shown that she and the DUP are continuing to ignore the views of the people of the north on Brexit," she said.
"The fact is that the majority of people from the north, from all backgrounds, voted to remain in the EU.
"When I meet people from all sectors from across the north it is clear they want to remain in the EU and that is what we are about and it is also what Arlene Foster's boss, Theresa May, should be focused on.
"The British government need to come up with credible proposals in the current Brexit negotiations, which would give effect to the views of the majority of people in the north."