Piers, 54, co-hosts the show with Susanna Reid and Richard often fills in for the presenter during half-term and over the summer.
Richard appeared on Tuesday's show to talk about taking part in Channel 4 reality show The Circle.
Towards the end of the chat, Piers said: "Actually you know what? Maybe Richard and I should present one of these together."
Richard said: "That is a great idea. Shake on it."
As the pair shook hands, Piers continued: "A male only GMB. Good Man Britain."
Susanna cut in: "You keep telling me that men are going out of fashion," to which Piers replied: "This is one way to fight back."
He continued: "Great Man Britain is coming. I think the viewers would love that. Me and Richard Madeley, the dream team."
Piers added: "Imagine if it worked and ratings spiked."
Susanna joked: "I'll take a couple of million and a payoff and you can get up at 3:50am, marvellous!"
Great Man Britain is coming. I think the viewers would love that.
Viewers loved the idea of Piers and Richard hosting together.
One person said on Twitter: "Really good show today. Good to see the nice side of Piers. We knew he had a heart really, and the Morgan-Madeley Bromance."
— Wildflowers (@8000wildflowers) October 8, 2019
Another wrote: "Yes you and Richard Madeley would be great together on @GMB."
A third tweeted: "@piersmorgan @EamonnHolmes Can we have you two and Dickie (Richard) on #Gmb dream team."
— Alison TT (@ALISONGBARMY) October 8, 2019
— Nelle (@SweetChariot30) October 8, 2019
During his interview, Richard spoke about rumours he could do Strictly Come Dancing.
Richard had to dance on The Circle for his co-stars and it involved an epic twerk.
However, the star admitted it was his "worst" moment and that's why he won't sign up to the BBC show.
He said: "That was my worst moment because I obviously cannot dance which is why I would never do Strictly. I’d be worse than James Cracknell," who was eliminated from this year's competition first.
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