Burton has been billed as one of the most promising young stoppers in England but is yet to make a senior appearance for the Shrews since signing a professional contract two years ago.
The 19-year-old was recently called up to an England Under-20s training camp after impressing scouts whilst in action for the League One club's academy side.
Burton is out of contract at New Meadow at the end of June, and according to the Mirror, Villa are ready to fight off Watford and Burnley to bring the teenager to Villa Park.
The goalkeeper's only first-team experience to date has come in the form of brief loan stints at National League sides Nuneaton and Southport midway through last season.
However, with Villa looking to cut costs and overhaul their squad in the summer as they look ahead to life in the Championship, Burton would be a good option to have between the posts.
Brad Guzan could leave the club in the summer after an inconsistent season if the club look to shift his significant wages, but Villa also have Mark Bunn, Jed Steer, and a number of academy goalkeepers including January signing Matija Sarkic on the books.
Meanwhile, Villa's relegation from the Premier League can be confirmed as soon as this weekend if they lose at home to Bournemouth and other results go against them.