The 18-year-old has been attracting a lot of interest from across Europe following a breakout season with Süper Lig side Trabzonspor.
Ömür made an impressive 32 appearances across all competition last season - almost five times the amount from the previous campaign - where he scored three goals and claimed six assists in the top flight.
It was understood that Leicester City were interested in bringing the teenager to the Premier League this summer and the club were even considering lodging a €20m offer for the versatile midfielder.
But HaberTurk have reported that Atlético are now the frontrunners for his signature because officials at the Wanda Metropolitano are prepared to pay roughly €25m for Ömür, who turns 19 later this month.
Atlético are also expected to face competition from domestic rivals Villarreal in the race for Ömür's signature this summer, while Leicester haven't given up hope of signing the midfielder.
The Turkey Under-21 international can play as a left midfielder of through the centre, but the majority of his time last season was spent down the right flank.
Ömür has largely had his best games against lesser opponents in the Turkish Süper Lig, although the 18-year-old did claim a vital assist as Trabzonspor secured an impressive 2-2 draw on the road to Beşiktaş.