In a petition run through Change.org, the former Downing Street comms director called on Dacre to "debate with me, or a professional broadcast interviewer, on TV, the attack on Ed Miliband’s father and the role of the Daily Mail in the UK’s national debate".
The move follows a clash between Miliband and the Daily Mail, which ran a story claiming his Marxist intellectual father Ralph Miliband "hated Britain".
The piece, followed by another that gave the Labour leader right to reply but repeated the accusation, was labelled by Miliband as "a lie".
It was reported this morning that Milband has now written to the proprietor of the Mail on Sunday, Lord Rothermere, after a family memorial service this weekend was gatecrashed by a reporter from the paper.
Campbell accused Dacre of being a "bully and coward" during an appearance on BBC’s Newsnight earlier this week.
He appeared alongside the paper’s deputy editor Jon Steafel, who defended the article and headline as "entirely justified".
In the petition he accuses the Mail of being "the worst of British values posing as the best" and calls on Dacre to "find the courage to debate in public".
Campbell was hired by Portland last year, joining CEO Tim Allan, his former deputy in Tony Blair’s office.
He worked as press secretary for Blair from 1994 and left Downing Street in 2003, before returning briefly to help the re-election campaigns of Blair in 2005 and Gordon Brown in 2010.
The Daily Mail has not yet responded to PRWeek's request for comment.
Note: Mail on Sunday editor Goergie Greig has released a statement apologising for the family memorial intrusion, stating: "The reporter was sent without my knowledge; it was a decision which was wrong." Two reporters have now been suspended and a full investigation is being carried out.