The 30-year-old former Crystal Palace winger impressed manager Darren Moore while training with the Baggies.
Talks have reportedly been ongoing for some time over a deal that would last until the end of the season.
Sako, who fractured his ankle in February, is a free agent after turning down the chance to stay at Palace, where he scored six goals in 19 appearances last season.
He has interest from other clubs but has been impressed by the progress being made by Moore and Albion's squad, who currently tops the Championship's scoring charts and lead the table by one point ahead of Leeds United.
Moore had asked Sako to get fitter ahead of signing the Mali international and said last week: "I can't say too much at the moment but if things develop we can talk more in detail. At the moment he's just training and getting fit.
"I will explain that if we do get round to signing him but at the moment he's not our player to comment on."