The 23-year-old was named Celtic's Player of the Year after ending the 2023-24 campaign as the club's joint-top scorer, netting 19 goals and registering 18 assists in 49 appearances across all competitions.
O'Riley has subsequently caught the eye of several clubs from across Europe, particularly Serie A side Atalanta BC, who recently had a second bid worth an initial £14.3m plus add-ons rejected by the Hoops, according to Sky Sports News.
Atalanta's Italian rivals Juventus and Roma have also been credited with an interest in the attacking midfielder, as have Atletico Madrid, Liverpool and Southampton, the latter of whom are allegedly ready to 'push the boat out' to sign the Celtic star for a Scottish-record transfer fee.
O'Riley is currently away with Celtic on their pre-season tour of the United States and he scored a brace as captain in a comfortable 4-0 friendly win over DC United on Sunday.
Rodgers relaxed over future of in-demand O'Riley
The Denmark international still has three years remaining on his contract at Parkhead, but Rodgers has admitted that Celtic are 'realistic' about O'Riley's situation and suggests the Hoops are a club who could be used as a stepping stone for players to join a bigger team.
Speaking about O'Riley to reporters ahead of Celtic's friendly against Manchester City in North Carolina on Wednesday, Rodgers said: "There's no update. We're fairly relaxed.
"He's working very well, he's very focused and, naturally, when you're a really good player and especially a young player who's as committed and professional as he is, he's going to acquire attention. He's working so well and there's nothing really to report."
"The people close to Celtic understand the model that works," Rodgers added. "Matt's been at the club for three years and developed into the player that he has.
Rodgers: 'Celtic is a club where you can move on'
"He knows the privilege it is to play for Celtic and, naturally, how it works is the players get that opportunity to move on. Celtic is a club where you can move on - but you don't have to. It's a very difficult club to move on from.
"With the wages that are on offer for players, that makes it a real challenge to compete at that level.
"What I've seen with Matt is that he's a first-class professional. He's a really humble guy who is devoted to his profession and he knows what he has here, so we're all quite relaxed.
"Every player has a value, but we see him as a really important player for us and while he is here, we'll continue to develop him and help him improve."
It is understood that an offer in the region of £30m, which would represent a Scottish-record sale, could persuade Celtic to cash in on O'Riley this summer.