After it begins its summer recess on Thursday, the House of Commons is not due to return until 1 September.
Commons Speaker Sir Lindsay Hoyle can recall Parliament during the recess if asked to by the government, once deciding whether it is in the public interest.
Shadow Commons leader Jesse Norman claimed the government “could not bear the idea of a new prime minister facing any scrutiny before September”.
He described the decision not to debate adding sitting days as a “total humiliation and embarrassment for the government”.
“It is the worst possible start for a new prime minister,” he told MPs on Tuesday.
“People across this country will see what has happened and they will conclude that this is a man who is frit, running scared of public scrutiny before he can even take office.”
Burnham, the former mayor of Greater Manchester who became the MP for Makerfield after winning a by-election last month, has been nominated by 349 Labour MPs to lead the party.
It is mathematically impossible for a rival to run against him, meaning he will become the Labour leader on Friday.
He is then set to replace Sir Keir Starmer as prime minister three days later, when the official transfer of power will take place.
Having twice run before to become Labour leader, his route to power is much clearer this time round.
As the sole candidate, Burnham will not need to go through the process of a summer vote of members of the Labour Party and affiliated trade union supporters.
The curtailed contest has prompted calls from some within Labour for Burnham to offer more details on policies he will look to implement in government.
He did not take questions from journalists after a speech last month and has conducted one media interview, with LBC, since confirming his intention to stand to replace Sir Keir after entering Parliament.
