It follows a High Court ruling in London that the government could not trigger Article 50 of the Lisbon Treaty - beginning formal exit negotiations with the EU - on its own.
DUP MP Nigel Dodds said his party will use their position to try to get a better deal for Northern Ireland in the event of Brexit.
Mr Durkan told the Sunday Politics programme that his party's position was clear and in line with the majority of people in Northern Ireland.
"If we get to a situation where parliament will be voting on the terms, I want to make one thing clear, we will be voting against Article 50 because that is our position and that is the position that the people of Northern Ireland have voted for," he said.
"That is also the position that our constituents have voted for, so the SDLP MPs will be able to say we are voting with our constituents and we are voting with the majority of people in Northern Ireland."
Mr Dodds said: "I expect to see as a result of the negotiations a deal which is in the mutual interests of the Irish Republic, of the other member EU states, of the United Kingdom and Northern Ireland."
Prime Minister Theresa May said the referendum - and existing ministerial powers - mean MPs do not need to vote, but campaigners called this unconstitutional.
Overall, the UK electorate voted to leave the EU, but in Northern Ireland, a majority of people (56%), voted to stay.
Thursday's High Court ruling was a defeat for the Westminster government, but it will now appeal the decision at the Supreme Court.
A further hearing is expected next month.
Northern Ireland's political parties remain sharply divided over Brexit ahead of a possible Westminster vote.
The SDLP MP Mark Durkan has said if MPs at Westminster are required to vote on whether the UK can start the process of leaving the EU, his party's MPs will vote against it.