Minchin announced on Tuesday that she was leaving the BBC's breakfast show later this year after 20 years of hosting.
Walker paid tribute to Minchin on Twitter after her announcement, and Morgan quickly joined in to joke he would be joining Walker on the BBC sofa.
In a whip-smart comeback, Walker quickly responded: “No thanks... wouldn’t want you storming off if Carol [Kirkwood] asked you something awkward.”
Morgan left GMB in March this year following an on-air row with weather presenter Alex Beresford over comments the former tabloid editor had made about Meghan Markle.
Walker had shared photos with his co-star, writing: “This feels like a break-up. What am I going to do without Louise?"
Watch: Louise Minchin announces BBC Breakfast departure
“Thank you for all the lovely messages after an emotional morning on #BBCBreakfast - I’m sure Louise will get to read them all later.
“My life will be poorer without her on the sofa. She’s the best.”
In response, Morgan wrote: “Stop blubbing Dan - I start Monday. BBC wanted the guy who beat you in the ratings, and the first same-sex male breakfast presenting team to appease the wokies."
Morgan and Walker have often shared jabs at each other over the rivalry between Good Morning Britain and BBC Breakfast particularly when it comes to their ratings.
Following Morgan’s abrupt exit from Good Morning Britain, Walker sent him a tweet that read: “Sorry to hear the news about your departure from GMB. If you’d like to come on the UK’s number 1 breakfast show to talk about it… we could make some space for you in the morning.”
In an interview with The Sunday Times shortly after, the BBC journalist claimed that there's "a decent bloke fighting to get out" of his long-time sparring partner.
He said: "Piers and I have had a friendly rivalry, but it has been good for BBC Breakfast and GMB.
"He likes to throw a few insults around, but underneath all the froth I think there is a decent bloke fighting to get out."
It seems Morgan did not take the comments kindly, and responded in his The Mail on Sunday column, stating: “Ironically, I’ve always thought that underneath all Dan’s faux saintly froth, there’s a nasty piece of work fighting to get out.”
Watch: Piers Morgan storms off Good Morning Britain after being challenged for Meghan Markle comments