Piers Morgan wants Madonna to "get down on bended knee" and apologize to him ... on TMZ -- all because of a mystery feud that blew up on the Internet this week.
You'll recall ... Piers publicly banned Madonna from his CNN show on Twitter Tuesday, writing, "Welcome to Twitter. You're still banned from my show. Love Piers X."
Madonna's manager Guy Oseary thumbed his nose in response, tweeting back Madonna had already been invited to Piers' UK show -- so Piers extended the ban worldwide.
Piers refused to elaborate on what's really behind the feud -- on his way out of an L.A. supermarket today -- telling us, "It's a private spat." (Piers also doesn't return his shopping cart to the cart corral in the parking lot ... but that's another issue.)
Piers added, "Madonna can come on [my] show, but she has to get on bended knee and apologize to me ... on TMZ."
But he probably won't get his wish -- because a source close to Madonna's management tells TMZ, "Piers is delusional for feuding with Madonna. He isn't even on Madonna's radar. The UK invitation speaks for itself."