The reports claim Szmodics is currently discussing personal terms with the Championship club.
Posh chairman Darragh MacAnthony has tweeted that Szmodics is available to play in the League One opener at Cheltenham on Saturday, but stopped short of denying a fee had been agreed.
"Sammie is available for selection for tomorrow,” MacAnthony said. “That’s our only focus!”
The Lancashire Telegraph has reported that there is a growing sense that the clubs will reach an agreement for Szmodics, with a deal looking increasingly likely but not imminent.
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The PT revealed earlier this month Posh had turned down three million-pound plus bids for Szmodics from Blackburn.
They paid Bristol City around a £1 million for Szmodics in August 2020 and he has since scored 23 goals in 85 appearances for the club. He scored four goals in 10 appearances when at loan at Posh in the 2019-20 season.