The Clarets are believed to have offered an initial £7.5m and up to £2.5m in bonuses for the 29-year-old, who is relegation-threatened Hull’s top goalscorer this season having netted seven times in the top flight.
West Ham and Middlesbrough are also understood to be in contention to sign the former Norwich and Leeds midfielder, but neither is thought to have matched the offer currently on the table from Burnley.
Snodgrass declined the chance to sign a new three-year deal at the Kcom Stadium last month, although Hull triggered a one-year extension which contracted him to the club until the summer of 2018.
Should Snodgrass move, it will be the second high-profile departure from Humberside during the transfer window after former Tottenham midfielder Jake Livermore joined West Brom for £10m last week.
Watford, meanwhile, have bolstered their striking options by signing former West Ham and Queens Park Rangers striker Mauro Zarate from Italian outfit Fiorentina for a fee believed to be in the region of £2.3m.
Zarate, 29, failed to start a Serie A match for Fiorentina this season and left England with a record of nine goals in 40 Premier League appearances but is set to be charged with pepping up a struggling forward line. Watford’s leading marksmen Troy Deeney and Etienne Capoue have just five goals apiece this term.
The signing of Zarate is set to intensify speculation regarding mis-firing forward Odion Ighalo, who has failed to replicate his goalscoring feats of last term and has been linked with a £17m move to West Brom.
The Hornets are also closing in on a loan deal for 22-year-old French forward M’Baye Niang, who has netted three goals in 18 Serie A appearances for AC Milan this campaign.
Championship side Aston Villa have strengthened their midfield with the signing of FC Basel’s Birkir Bjarnason – a member of the Iceland side which unceremoniously knocked England out of Euro 2016, for an undisclosed fee.