Ms Fox used to be a senior activist in the Revolutionary Communist Party (RCP). At the time of the 1993 bombing, which killed schoolboys Tim Parry and Johnathan Ball, an RCP newsletter stated that the party defended "the right of the Irish people to take whatever measures necessary in their struggle for freedom".
Tim's father Colin Parry said the peerage "offends me and many others deeply".
Ms Fox has said she does not condone violence but has never apologised for the position at the time.
Most recently she was a Brexit Party representative in Brussels for Nigel Farage’s Brexit Party.
Ms Rayner said Boris Johnson would be showing he "doesn't care about the victims and survivors of terrorism" if he does not block the peerage.
She added: "The awarding of a life peerage to someone who has repeatedly refused to apologise for her support of the heinous IRA bombing attack in Warrington in 1993 has rightly caused revulsion and real hurt both in Warrington and across our region.
"The Prime Minister is showing crass insensitivity to the families of those who lost their lives in 1993 and countless others whose lives were changed forever that day, and as a result of the IRA attack in Manchester city centre in 1996."
This is what happened when we asked Brexit Party candidate Claire Fox about her support for the 1993 Warrington bombings:
- ‘I don’t hold those views anymore because there is no war going on in Ireland anymore’
- Won’t apologise because ‘I didn’t do anything’ @GranadaReports pic.twitter.com/hw97UWaKvf— Hannah Miller (@hannahITV)May 19, 2019
Ms Fox has addressed the allegations in the past, saying: "Terrible things happened - mercifully a peaceful resolution has emerged following the Good Friday Agreement. I do not condone the use of violence."
A No 10 spokesman said: "Claire Fox has addressed her historic comments about the Troubles and acknowledged the pain that the families of the victims of terrorism have faced.
"She is not a Conservative peer, and her political views will differ from those of the Conservative Government."
The Standard has attempted to contact Ms Fox for comment.