The Ireland international will undergo a medical in the coming hours ahead of a switch to the Clarets.
The Championship leaders want to consolidate their 18-point cushion in the automatic promotion places by making some quality signings this month.
Obafemi will be challenging Jay Rodriguez and Nathan Tella for playing time at Turf Moor.
The 22-year-old has bagged three goals and an assist in 19 league games for the Swans this term.
Obafemi scored 12 goals and assisted three in 32 Championship games last season.
He has not played a game since action paused for the World Cup and will be raring to go.
Burnley have scored the most goals (55) in the division this season but are not resting on their laurels.
Tella and Rodriguez both have nine goals each, while Benson Manuel has seven goals.
Obafemi offers something different, with his electric pace and sheer power his major strengths.
The former Southampton striker also brings Premier League experience to the table from his time with the Saints.
It is almost guaranteed that Burnley will be playing in the English top flight next term.
Having Obafemi would also come in handy by the time they return to the division next season.
He scored four goals and assisted three in nine Premier League starts for Southampton.
Earning promotion with his soon-to-be new side will be top of his plans, and an official announcement is expected soon.
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