Transport secretary Chris Grayling has said he is “cautiously optimistic” about the parliamentary vote on Heathrow’s third runway.

The Conservative government is expecting significant support from the Labour party after the Unite union called on Labour MPs to vote in favour of the controversial third runway.

“I’m cautiously optimistic. It’s never over until it’s over and the vote has actually happened, but there is strong support from across the political spectrum for this,” Grayling told BBC Radio 4’s Today programme on Monday.

“It’s not usual for me to find myself campaigning on the same side as Len McCluskey of Unite, but he is right in arguing that this is a project that can make a real difference to Britain.”

Len McCluskey, the leader of Unite, said the third runway would be important in creating hundreds and thousands of new jobs in the UK.

When asked about the effects of the runway for climate change, Graylin said: “We’ve talked about climate change right the way through, and the airports commission, which started this work for us, looked very carefully at the issue of climate change.”

“The big difference that is being made in the aviation sector is technology. The new generation of aircraft are quieter, they are much more fuel-efficient.”

When questioned about who was paying for the runway, Grayling made clear that it would not be the taxpayer.

He said: “This is very clearly a private sector project. The taxpayer is not going to be paying for the expansion of Heathrow airport.”

“Some of it Heathrow will pay for, some of it the private sector will pay for, some of it the government will pay for. If you look at Crossrail, if you look at HS2, which are very much an important part of the package, these are things we’re doing already.”

Boris Johnson, who remains an opponent of the Heathrow expansion, is in Afganistan and will not be present for the vote.

The foreign secretary previously pledged to “lie down … in front of the bulldozers”.