The 28-year-old began training with the Baggies last week in the hope of proving his fitness and earning a deal until the end of the campaign.
A winger who is also able to play up front, Musa is currently a free agent meaning he is able to join a club outside the transfer window.
Back in 2016 the forward became Leicester's record signing when the Foxes paid £16.6million to sign him from CSKA Moscow.
But he struggled to make an impact in England and was loaned back to CSKA before going on to join Saudi Arabian side Al Nassr in a permanent deal.
Musa has been a free agent since leaving Pro League side in October and according to reports is eager to prove himself in the Premier League.
Speaking earlier this week, Nigeria boss Gernot Rohr said he was hopeful his captain would complete a move to The Hawthorns.
But the Express & Star understands Albion have decided against offering Musa a deal with the club feeling they already have stronger players in his preferred position out wide.
Grady Diangana, Robert Snodgrass, Kamil Grosicki, Matt Phillips, Kyle Edwards, Karlan Grant and Callum Robinson are all capable of playing on the wing – with Grant and Robinson also able to perform up front.
Meanwhile, the Express & Star understands Albion haven't received any recent approaches for Grosicki.
Widespread reports have suggested Legia Warsaw – who have until the end of this month to sign players – are interested in taking the winger back to Poland.
Speaking last month, Allardyce confirmed he is willing to listen to offers for Grosicki as Albion look to lower their wage bill.
Nottingham Forest, who came close to signing Grosicki in the summer, renewed their interest last month but the Poland international wasn't keen on a move to the City Ground.
It is believed Grosicki is hoping to secure a move to a European side this month in a bid to get more playing time ahead of the European Championships.