Worrall has made just one first-team appearance for the Baggies since joining the club from Bury two years ago.
The teenager spent a spell on loan at Accrington Stanley earlier this season and drops down to League Two again for a stint at Shrewsbury.
"He's a player who I have seen a lot of this season playing in their reserves and in pre-season," Shrews boss Paul Simpson told the club's official website.
Enthusiastic"He's an enthusiastic, fit footballer who has good technical ability and I just feel that we need a little burst of enthusiasm and some legs in there who will make a difference for us.
"There is no guarantee that he will go straight into the side, but I want to bring some people into the football club who I think can help us and David is a player who I feel could make a difference.
"He can play midfield or wide in the midfield, he's not a striker, but is definitely an attack-minded player.
"I want to look at him working around our first team and if he comes in and impresses as I'm hoping then we will look to extend it."