The party says MPs must have the chance to consider the decision to take the franchise from Virgin Trains and give it to FirstGroup for the next 13 years.
FirstGroup says it will bring in key improvements for passengers but critics fear it will not be able to afford the £5.5bn it is paying for the franchise.
The Department for Transport said it had "no reason" to delay the signing.
A spokesman indicated that the agreement was set to be signed on Tuesday despite Virgin's Sir Richard Branson offering on Sunday to run the service "for free" while a review was carried out.
The spokesman said: "We note the offer that one of the bidders appears to have made via the press.
"However, the winning bidder was decided by a fair and established process and no reason has been advanced to convince DfT not to sign the agreement."
Fierce lobbying
Aberdeen-based FirstGroup already operates a number of rail routes including Great Western and ScotRail.
The company, under the name First West Coast Limited, will take over the franchise from 9 December and is due to to operate the service until 2026.
The West Coast Mainline route serves 31 million passengers travelling between London, the West Midlands, the North West, North Wales and the central belt of Scotland.
Virgin has run the franchise since 1997, during which time passenger numbers have doubled.
FirstGroup has promised to introduce better wi-fi and food, more frequent trains and more seats, and to cut standard fares by 15%.
The firm said it would introduce 11 new 125mph six-car electric trains on the Birmingham to Glasgow route and provide more direct services between destinations.
But BBC deputy political editor James Landale said there was a fierce lobbying campaign, in the final hours before pen is put to paper, to put the decision on hold.
More than 100,000 people have signed an online petition calling for the decision to be reconsidered.
Labour's Maria Eagle, the shadow transport secretary, is urging the government to put the contract on hold until MPs return and have a chance to consider the deal.
And Labour's Louise Ellman, who chairs the House of Commons transport committee, has written to Transport Secretary Justine Greening asking her to delay signing the contract.
On Sunday, Sir Richard said that if reviewing the decision meant the December deadline had to be postponed, his company and Stagecoach would be willing to continue operating the railway lines while donating any profits to charity.
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