It is understood that Posh are currently in talks with the O’s over a move for the 21-year-old.
Kyprianou has a buy-out clause in his contract at Orient which another unnamed club are willing to pay.
The midfielder’s current deal at The Breyer Group Stadium is set to run until 2023.
Peterborough are believed to be pushing hard to match the fee that will be required to secure Kyprianou’s services.
A product of Orient’s academy, the midfielder made his debut for the club in 2019 during an EFL Trophy clash with Brighton & Hove End’s Under-21 side.
During the previous campaign, Kyprianou featured regularly for Orient as they finished 13th in the League Two standings.
The midfielder made 38 appearances for the O’s in the fourth-tier whilst he also represented the club in the FA Cup, the League Cup and the EFL Trophy.
Peterborough have already bolstered their midfield options this summer by swooping for Ben Thompson.
The 26-year-old signed an initial two-year deal with Posh on Tuesday after being released by Gillingham following the club’s relegation to League Two.
Quiz: Which club did Peterborough United sign these 24 players from in the 1990s?
If this report turns out to be correct, it will be interesting to see whether Peterborough will be able to secure the services of Kyprianou.
A promising talent, the midfielder managed to produce some encouraging performances for Orient in the fourth-tier during the previous term.
Having recorded a respectable average WhoScored match rating of 6.72 in League Two, Kyprianou will be keen to prove his worth in a higher division and thus could be willing to make the switch to Posh.
By learning from the guidance of Peterborough manager Grant McCann, the midfielder could go on to make considerable strides in terms of his development next season.