Any such inquiry could have serious implications for Starmer's future.
He has so far resisted pressure to quit over his decision to hire Mandelson but if found to have knowingly misled parliament his position would likely become untenable.
House of Commons Speaker Lindsay Hoyle said he had approved a request from opposition Conservative Party leader Kemi Badenoch for parliament to debate and vote on Tuesday on whether the Committee of Privileges should look into the matter.
Mandelson was fired by Starmer last September after it emerged his relationship with the late US sex offender Jeffrey Epstein was deeper than previously known.
That has raised doubts about Starmer's judgment in hiring him, exacerbated by the revelation that a security vetting body had described the appointment as a borderline case and that it was leaning against granting clearance - a decision foreign ministry officials overruled without telling the prime minister.
Starmer's Labour Party has a large majority in parliament, which could allow the government to instruct its MPs to vote down the launch of an inquiry.
On Monday the government published a letter sent in September from former cabinet secretary Chris Wormald, saying he had concluded "that appropriate processes were followed in both the appointment and withdrawal" of Mandelson as ambassador.
A spokesperson from Starmer's office described Badenoch's push for a vote as a "desperate political stunt" ahead of local elections due on May 7.
Hoyle said his decision to allow the vote should not be taken as an indicator of whether Starmer had done anything wrong or not.
If the United Kingdom parliament did vote in favour of an inquiry, the committee, made up of MPs from the three biggest parties, would examine whether Starmer's statements on Mandelson amount to knowingly or inadvertently misleading the House of Commons.
The focus of any such inquiry would be expected to fall on Starmer's statement that due process was followed when hiring Mandelson.
The committee previously found that former Conservative prime minister Boris Johnson had knowingly misled parliament over rule-breaking parties held during the COVID-19 pandemic.
Johnson had already stood down as prime minister by the time the report was published but he resigned from parliament altogether after seeing a draft copy of the findings.
