The 24-year-old has emerged as a leading target for Steve Bruce as the Hull boss continues to build a squad for a season in the Premier League.
Austin only has 12 months remaining on his contract at Burnley and it is understood that the two clubs are on the brink of agreeing a fee.
The former Swindon forward is then set to hold talks with the Tigers after previously being linked with the likes of Celtic and Southampton.
He has scored 45 goals for Burnley since his 2011 move from Swindon and he netted 28 times for the Clarets last season.
Bruce has been in the market for extra firepower following Hull's promotion to the top flight and has already seen a bid for Celtic striker Gary Hooper rejected.
The likes of Swansea's Luke Moore, West Bromwich Albion's Peter Odemwingie and South African forward Katlego Mphela have also been linked with moves to the KC Stadium.
The Tigers have so far signed Maynor Figueroa, Curtis Davies and Allan McGregor in addition to securing permanent deals for George Boyd and Ahmed Elmohamady.