In angry exchanges during PMQs on Wednesday, the prime minister accused the Labour leader of having “supported an IRA-condoning politician” when Jeremy Corbyn led the party.
Starmer, a former director of public prosecutions, demanded Johnson withdraw the comment.
“When the prime minister has worked with the intelligence and security forces prosecuting criminals and terrorists he can lecture me,” he said.
Following the exchanges, a spokesperson for Starmer said: “One thing we would remind the prime minister is that he has the power to block Claire Fox being nominated as a member of the House of Lords.
“So if he wants to take any action on this issue we suggest he does that.”
Fox was recently handed a peerage by Johnson. Before signing up to Nigel Farage’s Brexit Party in 2019, she was a leading of member of the Revolutionary Communist Party (RCP).
On March 20, 1993, an IRA bombing in Warrington killed Tim Parry, 12, and Jonathan Ball, three, and injured 56 others.
The RCP defended the bombing and the “the right of the Irish people to take whatever measures necessary in their struggle for freedom”.
Tim Parry’s father, Colin Parry, said at the time that offering a peerage to Fox “offends me and many others deeply”.
Fox has previously said she has the “greatest sympathy” for the Parry family but has not gone as far as apologising.
“My personal politics and views are well known and I have never sought to disguise them, though on this issue they have remained unaired for many years,” she said when asked about her political views at the time of the 2019 European elections.
“Whatever the rights and wrongs, the history of Ireland has been marked by tragedy and I acknowledge that without hesitation and with genuine feeling for all involved.”
The PMQs argument came after Starmer said Johnson’s government was responsible for “mess after mess”.
“U-turn after U-turn, it’s a fundamental issue of competence, God knows what’s going on, there’s no grip,” he said.
In reply, Johnson said: “This is a leader of the opposition who backed remaining in the EU and now is totally silent on the subject, now has performed a U-turn. He backed – in fact he still does Mr Speaker – this is a leader of the opposition who supported an IRA-condoning politician who wanted to get out of Nato and now says absolutely nothing.”
The prime minister’s comments were cut off by the speaker who called for order.