Everton's Goodison Park ground
Styles became the first player to play in a Football League fixture having been born in the 21st century upon making his Bury debut in May 2016.
The 16-year-old has since made eight further appearances for the Shakers, all of which have arrived since Boxing Day, and signed his first professional contract a fortnight ago.
According to the Liverpool Echo, however, that has done little to deter Everton, whose Under-23 scout Jamie Hoyland is said to have been present at recent Bury matches.
It is claimed that a number of unnamed Premier League sides are also chasing Styles' signature, although the Toffees' 2015 capture of former Shakers defender Matty Foulds could convince the two-assist midfielder to follow suit.
Everton have also snapped up several of the country's most promising talents in recent years, with John Stones, Mason Holgate and Ademola Lookman perhaps the most notable.
Everton's Ademola Lookman
Prior to signing for Bury, Styles was on the books of Everton's Premier League rivals Burnley.
The Toffees have also been credited with an interest in Newcastle United winger Owen Gallacher, whose great-uncle is former Manchester United manager Sir Alex Ferguson (Sunday People).