The 23-year-old is the subject of serious interest from across Europe this summer following a stellar 2023-24 campaign at Parkhead for the title-winning Bhoys.
O'Riley netted 18 goals and provided 13 assists in just 37 Scottish Premiership matches last term as Celtic pipped arch rivals Rangers to the Scottish crown.
Despite his best efforts, the midfielder was not selected for Denmark's Euro 2024 squad in Germany, where Kasper Hjulmand was successful at leading his charges to continental glory.
Returning to competitive action at the beginning of the new campaign, O'Riley picked up an assist during the 4-0 thrashing of Kilmarnock on August 4.
According to Sky Sports News, Celtic are holding firm at their valuation of the player, rejecting a fourth offer from Italian side Atalanta for the Dane.
The report claims that the Serie A outfit submitted a £20m bid for O'Riley, who is undoubtedly the standout star in the Scottish Premiership currently.
That bid is the same amount that Spanish powerhouses Atletico Madrid offered during the January window for the 23-year-old, highlighting how highly rated the player is across the continent.
Alongside the two aforementioned clubs, it is understood that Premier League sides Brighton & Hove Albion and Chelsea remain keen on a deal for O'Riley.
Sheffield United 'eyeing' Oxlade-Chamberlain deal
Moving onto the English Championship, The Sun are stating that Sheffield United are prepared to make a shock move for Besiktas man Alex Oxlade-Chamberlain.
The 35-time England international has spent the last 12 months at the Turkish giants, scoring four goals and setting up a teammate once in 40 appearances.
The efforts of Oxlade-Chamberlain were not enough to see his side to Super Lig glory, though, with Besiktas ending up in a lowly sixth position.
It is understood that the former Liverpool and Arsenal midfielder was eyeing up a move back to the riches and glamour of the Premier League.
However, it is said that Oxlade-Chamberlain is open to a switch to the second tier of English football, where Sheffield United are currently plying their trade.
It is believed that Blades boss Chris Wilder is keen on bringing the 30-year-old to Bramall Lane on a loan deal for the 2024-25 campaign.
Former Rangers manager Giovanni van Bronckhorst is now in charge at Besiktas, with the Dutchman willing to sanction a temporary transfer away for Oxlade-Chamberlain.
Everton 'considering' move for Belgian striker
Remaining in England, Football Insider are reporting that Everton are considering a surprise bid for Almeria striker Largie Ramazani.
The 23-year-old has been with the Spanish side since leaving the youth ranks of Manchester United in 2020 and has since made over 100 appearances.
Almeria were the whipping boys of La Liga last season and Ramazani's goalscoring record suffered as a result, scoring just three times in 29 league contests.
It is now being claimed that the second-tier club need to raise funds, with the sale of the Belgian a likely occurrence before the end of the summer window.
Sean Dyche's side have supposedly been alerted to the availability of Ramazani, who has experience of English football from his spell at Man United.
With a single year left on his current deal at Power Horse Stadium, the striker could be quite an affordable purchase for the Toffees this summer.