Posh's director of football Barry Fry told BBC Radio Cambridgeshire: "He [Mackail-Smith] feels a bit upset about the fact we won't allow him to grow.
"But he's been in this position before and he'll roll his sleeves up and run his heart out for the club."
Fry said the offer from Norwich came in before Saturday's 6-0 win over Carlisle.
"We had a bid last week and I spoke to Paul [Lambert] and David [McNally] from Norwich," he said.
"I put it to my chairman and discussed it with my manager, but the chairman has the last say on everything and he's rejected their proposal.
"[Mackail-Smith] knew about it before the game on Saturday, but he went out, he made the first goal, scored the second and got brought down for the penalty, despite being ill."
Since signing from Dagenham and Redbridge in January 2007, Mackail-Smith has scored 90 goals in 198 appearances, and regularly drawn plaudits for his committed performances on the pitch.
Posh have previously rejected bids from Hull and Reading, but Fry admitted the striker was unlikely to be at Peterborough beyond this season.
"If you're offered a better job with far, far more money and playing at the top level, it's every footballer's dream," he said.
"We've stopped him fulfilling that dream five or six times.
"I'm sure in the summer there'll be 20 clubs after him and the chairman has gone on record to say he won't sign another contract so he'll be sold in the summer.
"Sometimes you've got to say, enough's enough, you've done brilliantly for the club and you go with our blessing."
Source: BBC Sport
Source: BBC Sport