'The Daily Mail' reports that the players has already agreed personal terms with Sean Dyche's side and is expected to return from holiday to undergo a medical withing the next 48 hours, with a view to joining up with his new team-mates on Monday.
The contract on offer is believed to be a two-year deal with an option for a further year.
Burnley have been chasing the player for some time now and had an initial £1m offer rejected last month.
Now a compromise has been reached it is expected that the deal will move quickly.
The 33-year-old has scored 62 goals in 271 games for Stoke since joining from Ipswich in 2010, though he was little more than a bit-part player in the Potteries last season, starting just 13 Premier League games.
Burnley are also expected to complete the signing of Leeds' Charlie Taylor on a free transfer.
The club have agreed terms with the player though the deal is being held up by the compensation package that Leeds would be due.
Burnley are believed to have offered £5m for the player but Leeds are confident they would get more by waiting for a tribunal.
Either way the deal is expected to go ahead.