The 25-year-old arrived at Elland Road on Thursday hoping to finalising a three-month deal.
However, chief operating officer Shaun Harvey told BBC Radio Leeds: "We have been unable to do the deal in time for Saturday's game at Burnley.
"But we anticipate that it will be concluded in the early part of next week," he confirmed.
Hulse started his career at Crewe where he scored 45 goals in 97 appearances before joining the Baggies for £750k in the summer of 2003.
He was top scorer last season with 13 goals but has made only seven appearances for them this season following promotion to the Premiership.