The Clarets have turned their attention to the 29-year-old after they saw two bids knocked back for Brady.
Snodgrass has been Hull's standout performer in a difficult season on Humberside, scoring nine goals in 24 games and is attracting interest from a number of clubs.
Besides Burnley, Middlesbrough, West Ham and Championship side Newcastle are thought to be interested in the Scotland international.
Sean Dyche is keen to add more options on the flanks, to ease the workload of relentless duo George Boyd and Scott Arfield.
Snodgrass has 18 months to run on his contract at City, as the club decided to exercise an option to extend his current deal by a further year.