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 providing 18 assists in 49 appearances across all competitions.
O'Riley has subsequently been the subject of much speculation throughout this summer and has been linked with a number of clubs across Europe.
The Denmark international was wanted by Atletico Madrid in January, but a £20m move to La Liga ultimately failed to materialise and the Spanish club have since turned their attention to other targets, including Conor Gallagher.
Atalanta BC have been pushing to sign O'Riley over the last month and have been left frustrated in negotiations with Celtic, who have turned down five bids from the Serie A club, including their highest offer of £21.5m plus add-ons.
Brighton set to win race for in-demand O'Riley
Premier League clubs including Chelsea and Southampton have also expressed an interest in O'Riley, but Brighton now appear to have moved to the front of the queue for his signature.
According to journalist Fabrizio Romano, Brighton have reached an agreement with Celtic to sign the midfielder for a fee which could rise to £30m including add-ons.
It is understood that the move will set a new record transfer fee received by a Scottish club, surpassing Celtic's £25m sales of Jota and Kieran Tierney.
A separate report from Sky Sports News claims that the finer details of the deal are still being discussed, but there is 'confidence on all sides' that the transfer will be completed this weekend.
O'Riley, who is said to have agreed a five-year deal with Brighton, trained with his Celtic teammates on Friday ahead of their visit to St Mirren this weekend, and manager Brendan Rodgers was quizzed about the midfielder's future during a pre-match press conference.
Rodgers coy on O'Riley future prior to Brighton agreement
"Matt trained this morning as he always does. He is the consummate professional and has been brilliant," Rodgers told reporters prior to Brighton's alleged agreement. "There have been a number of offers gone in for Matt but there is nothing new to report or add to that either."
Asked whether there comes a time when offers for players will be too late to be considered, Rodgers added: "Yes. For any player, it is not going to go right down to the wire. We have got a big week next week after our game at the weekend.
"We will be preparing for our last game in this block of games. I wouldn't want that to be interrupted at all. We are pretty clear on where we are and what it is we need to do."
Brighton are also understood to have reached an agreement with Fenerbahce to sign Turkish international left-back Ferdi Kadioglu for £30m (£25.4m).
Fabian Hurzeler has already been backed in the summer transfer window, with the Seagulls spending in excess of £140m on seven new arrivals and breaking their transfer record twice to sign Yankuba Minteh and then Georginio Rutter, the latter arriving from Leeds United for £40m.