Football League clubs have until the close of play on Thursday to bolster their squads ahead of the season's run-in and Shrews boss Graham Turner is keen to bring in Hurst and Asante.
Hurst enjoyed a spell on loan at Shrewsbury in 2011 making seven appearances for the club during a two-month loan stint at Greenhous Meadow.
Full-back Hurst spent a month on loan at Birmingham earlier this season and West Brom are happy to farm him out again so that he can some valuable action under his belt.
Asante is well down the pecking order in Lee Clark's plans at St Andrew's this season so he is happy to let him go out on loan to work on his fitness.
Shrewsbury hope to land the duo in time for their Good Friday clash with Carlisle as they battle for survival in League One.