The 21-year-old midfielder is the subject of a £1.5million transfer which would see him swap English League One for the Scottish Premiership.
According to The Athletic, Celtic will win the race for his signature ahead of clubs from England and around Europe with the deal to be concluded by the end of the week.
Both Swansea City and Blackburn Rovers have been strongly linked with the player. Swans boss Chris Martin, the former Rangers and Scotland defender, was O’Riley's boss at MK Dons before making the move to Wales.
Swansea are nearing a deal to bring Hannes Wolf, the Borussia Monchengladbach winger who has been linked with a move to Celtic, in on loan this month.
The former England youth international has scored ten goals in 54 appearances for MK Dons since joining from Fulham last January. He is currently ninth in League One for goals and assists with a combined 12.
A left-footer, he has largely played centrally but is adept at taking up different positions in the midfield, especially looking to pick up space behind the opposition's midfield.
Only former Scotland international Barry Bannan averages more through balls than O’Riley this season in the third tier.
Due to the deal which took him to MK Dons, Fulham will be due a significant sell-on clause.
During his time with the Cottagers as an academy player both Arsenal and Manchester City were rumoured to be keen on the player when he was a teenager.