MPs are to push forward plans to give new mothers and fathers the right to nominate a party colleague to vote on their behalf in parliament.

The Labour MP Harriet Harman said she expected a backbench business debate in October, where MPs could vote on plans for MPs to be able to register a proxy vote when they are absent with a newborn.

Harman said the House of Commons was changing for the better, with a new demographic of young female MPs who wanted to have a family as well as a political career, which required a change in the rules. Currently MPs can only vote in person, or are registered as absent.

The proposal has cross-party backing, including from Lib Dem deputy leader Jo Swinson, Conservative select committee chair Maria Miller and the SNP’s Hannah Bardell.

Swinson, who has recently had her second child, was at the centre of the row during the summer during a tight Brexit vote where the Conservative chair Brandon Lewis broke a so-called pairing agreement, where rival parties agree to have MPs abstain in order to cancel out opposition MPs who are ill, new parents or unavoidably absent.

MPs will have a general debate on the issue of proxy voting in parliament on Thursday and a government motion on parental leave is also expected, with the backing of Andrea Leadsom, the leader of the House of Commons.

A government source said they were hoping to secure Downing Street support for the change, but said it would come down to timetabling as to whether Leadsom or Harman’s proposal was put before MPs first.

Harman said she expected that MPs would be able to vote on her motion, based on recommendations from a report by the Commons procedure committee, at a backbench business debate in October.

MPs have said it is vital the issue is resolved swiftly, given the number of MPs who are new parents or expecting. Labour has three MPs who are pregnant or on maternity leave – Laura Pidcock, Cat Smith and Holly Lynch – and the Conservative MPs Andrea Jenkyns and Chloe Smith have recently given birth.

Emma Reynolds, the Labour MP who gave birth four days before the election last year, said the pairing system was “not fit for purpose” and meant new mothers could be vulnerable to allegations of laziness if their voting record appeared to be poor.

“Last year, a national newspaper accused me of having the second-worst voting record in parliament without referring to the fact I was on maternity leave,” she said. “One of my other colleagues in a similar situation was branded one of ‘Britain’s laziest MPs’. New mums shouldn’t have to face this reputational damage.

“I hope that these proposals for proxy voting will go through quickly so that MPs can continue to represent their constituents while they are on maternity leave, paternity leave or shared parental leave.

“Passing these reforms would send out an important signal to young women and men who are considering a career as an MP and who would also like to start a family.”