Jeremy Corbyn has tabled a motion of no confidence in Theresa May, after she said MPs would not vote on her Brexit deal until the week of 14 January.
The PM had delayed the vote from last week, admitting she was set to lose.
Labour leader Mr Corbyn said on Monday it was unacceptable for MPs to wait a month to vote, adding the PM had led the UK into a “national crisis”.
But No 10 sources told the BBC the government would not make time for the no-confidence vote.
Ministers would not “go along with silly political games”, they added.
Mr Corbyn tabled the motion calling on MPs to declare they have “no confidence in the prime minister due to her failure to allow the House of Commons to have a meaningful vote straightaway” on the Brexit deal.
The motion focuses on Mrs May personally, rather than the government.
BBC political editor Laura Kuenssberg said the motion could have been embarrassing for Mrs May, but as things stood, ministers would not allow time for it to be debated.
She said No 10 had effectively “batted the ball back to Labour to see if they have the guts” to call for a vote of no confidence in the government as a whole.
Unlike a vote targeting the PM, a motion of no confidence in the government could bring about an early general election if it is supported by a majority of MPs.