The winger has spent the last three years at Crystal Palace, but the Mali international rejected the offer of a new contract from the Eagles and has been without a club over the summer.
The two parties allegedly came close to being reunited in recent weeks, but according to Midlands journalist Tim Nash, Sako will soon link up with the Baggies.
The transfer - should it be completed - will be considered controversial in the Black Country, with Sako having previously spent a successful three years with West Brom's fierce rivals Wolverhampton Wanderers.
The 30-year-old scored a total of 37 goals in 124 appearances for Wolves before opting to leave in 2015 due to the club failing to secure promotion to the Premier League.
Sako netted five times from 15 starts and 28 substitute outings for Palace in the top flight.