The 21-year-old has emerged as one of a number of candidates on Moyes's midfield shortlist ahead of the January window, with Inter's Fredy Guarin also subject to heavy interest.
Moyes has sent scouts to watch Guarin in action but elected to monitor Koke first-hand and was given a VIP seat at the Calderon, where he spoke to the club's sporting director, Jose Luis Caminero.
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"We spoke, but he didn't tell me exactly why he had come..."
Koke played the 90 minutes as Diego Simeone's men, who had already qualified for the last 16, beat Porto 2-0 thanks to goals from Raul Garcia and Diego Costa.
The 21-year-old has scored 10 goals in 134 games for Atletico and has also represented Spain on seven occasions after making his debut for La Roja earlier this year.
The midfielder recently put pen to paper on a new deal at the Calderon to extend his current contract to 2018 and told Canal Plus: "I didn't know anything [about Moyes being here] but I'm not concerned [with any of that]. I renewed in the summer."
United have been tracking Koke for several months and believe that he fits their profile as a potential signing due to his age, outstanding technique and versatility, with the Madrid-born youngster able to cover a number of different positions across the midfield and attack.