With the January window about to open, it is thought that the Clarets will up their offer for the 17-year-old Oldham Athletic product, with the Oldham Times reporting that a bid of around £200,000 was turned down in the summer by the Lemsagam brothers.
But Burnley have kept a close eye on the Republic of Ireland Under 19 international, who is also believed to have caught the attention of a number of Premier Leagu rivals, including Manchester City and Everton.
Vaughan made his first team debut in February under Harry Kewell, as a late substitute in a defeat at home to Barrow, and made five further appearances from the bench last term, before signing a two-year contract, with the club taking an option for another 12 months.
City are understood to have had a look at the youngster at their Etihad Campus following the end of the EFL season, training with the Blues' Under 18s.
He wasn't involved in Burnley's pre-season trip to Boundary Park, as he observed Covid protocols, but he has impressed this season, making his first start in the Carabao Cup second round penalty shoot-out triumph over Accrington Stanley, before scoring his first senior goals against Harrogate in the league, and the winner against Sunderland in the Papa John’s Trophy.