Dale boss Keith Hill says the League One side have offered a "tremendous amount of money" in an attempt to secure Hogan's services.
But the former Hyde forward, who has scored 12 goals in 26 appearances this campaign, wants to stay at Rochdale.
Hill told the club's official website: "The fact of the matter is that we're a business as well as football team. It was obvious to me that we were probably going to start getting enquiries for Scott, but with respect to Peterborough they've put their money where their mouth is and have come up with a tremendous amount of money.
"With respect to showing their intent, and from a football club's perspective, I 100% cannot complain with the football club [for accepting the bid] from a business aspect.
"It's a tremendous amount of money and while I've been at this football club, including during my previous regime, we didn't receive any bids of this amount for any of the players.
"Bravery and respect must go to Peterborough because it's a decent amount of money that we have had to accept, but then it was over to Scotty.
"Scott was then given the opportunity to speak with Peterborough and at this moment in time he's adamant that he wants to stay and I'm over the moon with that.
"It shows his intent and his trust in myself, the football club and the players. He's enjoying his football and at this moment in time Scott remains a Rochdale football player and that is absolutely superb news for the supporters.
"But everyone has got to understand that there is a business angle to provide the funds and the financial security for the future.
"As a football club we could not turn down the advances of Peterborough, the one person who can do is Scott and at this moment in time I'm glad that he is doing."