The offer arrived on Thursday morning as MacAnthony was at the airport waiting to board a flight to America for a family break.
The Irish businessman refused to reveal the identity of the player concerned although he did say it was not for highly-rated striker Britt Assombalonga, who scored 33 goals this season as Posh lifted the Johnstone's Paint Trophy at Wembley and reached the League One play-offs.
"Football still chasing me even when I'm trying to get a break. At airport about to board a plane and phone rings. We want such-and-such (not Britt Assombalonga) and I'm like, sorry nope, he's not for sale," said MacAnthony via his Twitter account.
"We will give you £2million today. I said no thanks but appreciate the call!
"Calls like that give me great confidence that our policy as a club is the right one and the only one. I'm now starting to get excited again for refresh/rebuild."