The Clarets have been chasing Obafemi, but there was doubt over whether they would press forward with their interest after signing Lyle Foster from Westerlo.
The interest remains though and Burnley are now rapidly closing in on a deal.
A fee has been agreed between Burnley and Swansea for Obafemi and the player is set to undergo a medical with the Clarets ahead of completing the move.
Burnley are in a strong position at the top of the Championship table, but Vincent Kompany still wants to add Obafemi to the ranks.
Obafemi has been a regular for Swansea so far this season, clocking 19 Championship appearances with a return of three goals.
Kompany got an up-close look at the 22-year-old earlier this season when Burnley visited Swansea and ran out 4-0 winners.
The Clarets top the Championship table with 62 points from 28 games, giving them a whopping 18-point advantage over third placed Watford.
