Watch: Piers Morgan doubtful over Alistair Campbell hosting GMB
This week former New Labour spin doctor Campbell, 63, made his debut as Susanna Reid's new co-anchor after Morgan, 56, quit the ITV breakfast show in March over comments he made about Meghan Markle's interview with Oprah Winfrey.
The exchange began with a tweet from Morgan about celebrities promoting Mental Health Awareness Week.
The former Britain's Got Talent judge wrote: "I’m curious to know just how much money famous ‘mental health campaigners’ are making from constantly preaching about it & exploiting their myriad ‘problems’ - and what percentage of that money they give to mental health charities. Celebrity victimhood is a lucrative business."
Campbell – who has battled depression – seemed to think the dig was directed at him, and replied: "If you mean me @piersmorgan - @schooltruth [his civil partner journalist Fiona Millar] and I have a charity account which takes a share of my earnings, which go to causes we believe in, mainly mental health and education for poorer children. I do paid speaking gigs and often ask for fees to be paid to charity.
"I have made money from books and documentaries about mental health, (I do have to make a living) and in all cases some of the proceeds go to charity. I am an ambassador for @mind_ @Rethink_ a patron of @MaytreeRespite and Global Ambassador for Australians for Mental Health.
"I’m also involved with MH charities in Ireland + Scotland. I have sometimes been paid to advise organisations on mental health in the workplace and again always will make donation to sector organisation."
Morgan hit back: "No, I was talking about celebrities."
The former government press secretary and the ex-tabloid editor have had several public disagreements over the years.
Campbell was a leading government adviser during the 2003 Iraq war, which Morgan – then editor of the Daily Mirror - was strongly opposed to.
Morgan was then sacked from the Mirror in 2004 after publishing photographs of British soldiers abusing Iraqi prisoners that turned out to be fake.
The pair were due to take their differences into the boxing ring last year as part of a charity match for Sport Relief.
But Morgan pulled out, saying: "I mulled it over, would it be a good or bad career move if I was to kill Alastair Campbell in a boxing ring?
"I took an honest human view. I decided to spare Alastair Campbell's life."
Campbell also replaced Morgan in 2014 when he took over his role as celebrity interviewer for men's magazine GQ.
Watch: Piers Morgan has voiced doubts that Alastair Campbell is a suitable presenter for GMB