Posh paid a club-record fee of £1.2million to sign Barnett from the Red Devils last season.
The 27-year-old was made available for transfer by Posh this week after a breach of club discipline.
Crawley boss Richie Barker admits bringing Barnett back to Broadfield is something he often talks about, but he feels the club would probably not be able to afford to do it.
"Tyrone has been here before and I know he is someone the fans would like to see back, I barely go a day without someone talking to me about him," Barker told the Crawley News.
"But Peterborough paid a fair amount for him and I should imagine they would like to see some of that back, even if he has been transfer-listed. The finances at Crawley aren't a bottomless pit, we can't be going out and spending half a million pounds on someone."