O'Riley has endured a difficult start to the season, missing time through illness and struggling for minutes under manager Fabian Hürzeler, and he is not expected to be involved in Brighton's squad for Sunday's game against Aston Villa.
According to Sky Sports, that absence is connected to transfer activity surrounding the 24-year-old rather than fitness or form.
O'Riley is said to be weighing his options, with a return to French football emerging as a genuine possibility. Lyon are among the clubs to have shown interest, according to Sky, and the Ligue 1 side are exploring both a loan and a permanent deal for the midfielder.
Lyon already have two Danish midfielders on their books in Noah Nartey and Mads Bidstrup, which could ease O'Riley's integration should the move go ahead. He has previous experience of French football, having spent time on loan at Marseille earlier in his career.
Beyond Lyon, Sky reports that O'Riley is also attracting interest from several other clubs across Europe's top five leagues, giving him a range of options to consider before the window closes.
The midfielder has made 29 appearances for Brighton so far and is under contract at Falmer Stadium until the summer of 2029, meaning any departure this window would likely require a fee, whether structured as a loan with obligations or a straightforward permanent transfer.
O'Riley joined Brighton from Celtic in a deal worth £25 million in 2024, arriving with a reputation as one of Scottish football's most productive midfielders after a string of strong seasons in Glasgow.
His time on the south coast has been disrupted throughout, starting with an ankle injury on his debut, and he was sent on loan to Marseille last season before being recalled by Brighton in January.
Brighton have also brought in fresh competition for places in that area of the pitch this summer, further squeezing O'Riley's opportunities and reinforcing the sense that a temporary or permanent exit could suit both player and club at this stage of the window.