The pair were discussing the former IRA chief’s legacy after he passed away aged 66, on Tuesday, March 21.
Mr McGuinness was a controversial figure after leading Derry’s IRA faction before playing a key role in the Good Friday Agreement, in 1998, which helped bring an end to the Troubles.
He went on to act as a senior figure in Sinn Fein, as he served as deputy first minister of Northern Ireland from May 2007 to January 2017.
Mr Campbell, a close ally to Tony Blair, who helped broker the peace deal, claimed Mr McGuinness “was a big part” of the peace process despite his past in the IRA.
Alastair Campbell's tributes to Martin McGuinness left a IRA victim's sister fuming
It is unbelievable that Martin McGuinness is being lauded and applauded as a statesman
Speaking on Channel 4 News, he said: “I can understand why it is impossible for people for to forgive or forget.
“But I think you also have to look at whether Northern Ireland is now a better place than it was and acknowledge – friend or foe – that Martin McGuinness and they journey he made as an individual is a massive part of that.
“If we never forgive, if we let go of the hatred that exists between people, then you never make progress. It is often about individuals and that is why I think Martin McGuinness, and what he did, is fundamental to the progress that’s been made.”
Ms Hambleton lost her sister Maxine when IRA terrorists targeted two Birmingham city centre pubs – the Mulberry Bush and the Tavern in the Town – in November 1971.
Angered by Mr Campbell’s comments, she blasted: “It is unbelievable that he is being lauded and applauded as a statesman.
Get Quotes on Home Insurance
“For Alastair Campbell to say he understands why he feels the way we do is also unbelievable. Mr Campbell, you have not lost a loved one where you as a father have to go and identify your son or daughter, where the sheets are pulled back you can barely recognise what is left of them.
“Please, please do not patronise us. As far as Martin McGuinness is concerned, if he was a statesman, a statesman has integrity, courage, bravery – all these things he lacked.”
Earlier on Tuesday, Mr Hambleton, a founder of Justice 4 the 21, told talkRADIO: “Today is a day we should be remembering the thousands of people who were murdered in cold blood.
Martin McGuinness' life in pictures Tue, March 21, 2017
The former deputy first minister of Northern Ireland, has passed away aged 66
AFP/Getty Images 1 of 13
Martin McGuinness and Tony Blair look on at Stormont Parliament Buildings in Belfast in 2007
“This man is being heralded as a peacemaker, almost as a god, when in actual fact he was just a cog in a wheel of a big project to bring about peace in Northern Ireland.
“Nobody is immortal and this catches up with everybody, however, he was fortunate, he lived a full life – thousands of people didn’t.”
Mr McGuinness’ death was announced early on Tuesday morning after battling a rare heart condition.
The 66-year-old died in the Altnagelvin Hospital in Derry with his family by his bedside.