Sir Keir Starmer is facing growing pressure to axe the two-child benefit limit.
The Scottish National Party joined Labour Leftwingers in pushing for a Commons vote on the controversial restriction.
Liberal Democrat MPs have also backed scrapping the cap.
John McDonnell, Labour MP for Hayes and Harlington, said last week that he is ready to table an amendment to the Finance Bill, which would be debated in Parliament after the Budget in the autumn, to ditch this benefit limit.
But the SNP is proposing a similar move to the King’s Speech on Wednesday.
The party’s Westminster leader Stephen Flynn has written to Anas Sarwar, the Scottish Labour leader, urging him to instruct his party's MPs in Scotland to support the SNP move.
Many Labour MPs privately, and some publicly, backing axing the restriction.
The cap, introduced by the Conservatives in 2017, prevents parents claiming Universal Credit or child tax credits for a third child, except in very limited circumstances.
The SNP Westminster leader will table an amendment to the King's Speech, which will be made on Wednesday setting out the Government's legislative agenda.
MPs then have the opportunity to debate the contents of the speech in the following days, at which point they can lay amendments to it.
Power over which amendments are selected is in the gift of Commons Speaker Sir Lindsay Hoyle.
In his party's first major intervention since the General Election, Mr Flynn said the two child cap is "pushing thousands of Scottish children into poverty" and ending it is "the bare minimum" required of the new Government.
His letter to the Scottish Labour leader claimed the "Tory two child cap became the Labour Party two child cap" once Sir Keir stepped through the door of Downing Street.
Sir Keir said last month that he would ditch the two-child limit “in an ideal world” but added that “we haven’t got the resources to do it at the moment”.
Figures published last week by the Department for Work and Pensions showed there were 1.6 million children living in households affected by the cap as of April this year, up from 1.5 million to April 2023.
Of these, 52 per cent of children were in households with three children, 29 per cent in households with four children, and 19 per cent in households with five or more children.
The Resolution Foundation has said that abolishing the two-child limit would cost the Government somewhere between £2.5 billion and £3.6 billion in 2024/25, but that such costs are "low compared to the harm that the policy causes".
