The 28-year-old has been unattached since quitting Al Nassr in October 2020.
West Brom has requested a UK visa for Musa and he is expected to come in this Wednesday for medical tests. Should all go well, he will sign until the end of the season.
The captain who has been capped 95 times by Nigeria, experienced a stalled career since he left CSKA Moscow in 2016.
Having scored 55 goals in 167 appearances for the Russians, Musa joined Leicester City in the summer of that year, however, things did not work out at the King Power Stadium, as the African made just seven league starts in 21 outings.
He joined after the Foxes had claimed the Premier League title and hoped to be part of something special with the club but he was loaned back to CSKA for the second half of the 2017/18 campaign.
He was then sold to Al Nassr in January 2019 and played 62 times in all competitions before being released.
Allardyce wants to bolster his attacking options and believes Musa’s pace and power can add another dimension to his struggling side as they look to avoid relegation. West Brom is still in the market scouting for targets across Europe.