HITC Sport understands Sunderland are eyeing the Championship winger.
Moyes was an interested observer for Forest's Championship game against Brentford on Tuesday with a scouting mission on Burke the reason for his attendance at Griffin Park.
However, Moyes was disappointed as Burke started the game on the bench and only came on after 66 minutes of the 1-0 defeat to Brentford.
A number of Premier League clubs are understood to be tracking flying winger Burke, with Forest facing a battle to keep hold of the teenager before the close of the transfer window.
Burke impressed for Forest last season and he has again the caught the eye so far this term, scoring in the opening-day victory over Burton Albion as well as showing up well in the defeat to Brighton last week.
Sunderland boss Moyes is thought to be a big fan of his fellow Scot Burke and is considering making a move for the 19-year-old.
Moyes is keen to bolster his attacking options at the Stadium of Light and he sees Burke as someone who could prove a valuable addition to his squad for the foreseeable future.