Striker Holt hobbled off during the closing stages of last weekend's 2-1 home win over Wigan, which extended the Canaries' unbeaten run in the Premier League to 10 matches.
The 31-year-old, who has netted four goals this season, revealed on Twitter earlier this week scans had shown the problem was "not as bad as first thought".
City boss Chris Hughton will leave a decision on Norwich's talisman until as late as possible.
"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 told a press conference.
"If he did miss West Brom, we would certainly hope he is fit enough for the rest of the Christmas period."
Despite concerns over Holt, there was positive news this week on the recovery of Harry Kane.
The on-loan Tottenham striker, 19, stepped up his rehabilitation from a broken metatarsal with a run-out for the under-21s, scoring twice in the 3-1 win over West Ham.
Kane had made just two appearances for Norwich before he suffered the injury against Doncaster in the Capital One Cup at the end of September.
Hughton, though, revealed the England Under-19 forward could come into contention ahead of the hectic festive schedule.
He said: "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.
"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."
Norwich - who bounced back from their home Capital One Cup quarter-final defeat by Aston Villa - have not lost in the Premier League since going down 4-1 at Chelsea on October 6.
It is a remarkable run of form, second only to leaders Manchester United, which has seen the Canaries fly up the table to put daylight between themselves and the relegation zone.
Hughton, though, knows his team face a stern test of their new-found resilience over the next fortnight - which will see both Chelsea and champions Manchester City visit Carrow Road and then a trip to West Ham on New Year's Day.
"It has to be very much one game at a time," the Norwich manager said.
"Once you get through that first game in the sequence of four, you see how you have come through that in terms of fatigue or injury or result and the next squad will generally be based on what you have seen.
"It is the same for everybody. It is not like we are playing a series of games and the rest are not.
"We have a real competitive squad at the moment and are fairly light on injuries.
"This is a time when everybody wants to be involved, but we have two tough away games and two games at home that do not get any tougher."