The striker limped off towards the end of last weekend's 2-1 win over Wigan, which extended Norwich's unbeaten run in the Premier League to 10 games.
"The good news is it is certainly not anywhere near as bad as either it could be or as we feared it might be - but I would have to say Grant is touch and go for the weekend," Hughton said.
"If he did miss West Brom, we would certainly hope he is fit enough for the rest of the Christmas period."
If Holt is ruled out, his place in the squad could be taken by fit-again Harry Kane.
The on-loan Spurs striker, 19, scored two goals in the Canaries' U21s victory over West Ham in midweek as he returns from a broken metatarsal.
"Harry is certainly there or thereabouts. The decision we have to make on Harry is the fact he has been out for such a long time," Hughton said.
"Harry has done a fair bit of work, but he will be lacking in some areas, which is normal, but the good news is the injury is fine.
"It is good to have him back. If he was involved (at West Brom) it would be based on the fact he has done enough. We don't want him to break down."