The Baggies are ready to activate a £4.5million clause to buy the 26-year-old, who is on loan from Al Ahly, next month.
Hegazi has impressed since joining from the Egyptian Premier League champions in July and has been linked with Liverpool.
Boss Alan Pardew said: “We’re pretty much down the road on that one.
“I can’t confirm anything but if we can then we’ll do that. He’s been brilliant for me.
“I’ve only been here for three games but he’s been terrific. He has fantastic attributes as a centre-half and I think he’s learning.
“Sometimes he gets a bit too tight but he’s learning to keep his distance. He’s been excellent.”
Albion host Manchester United on Sunday after a 0-0 draw at Liverpool on Wednesday lifted them out of the Premier League relegation zone.
The point still leaves a club record run of 16 games without a win but Pardew believes they have gained momentum from Anfield.
“I’m not sure there’s many teams that would have got a clean sheet at Liverpool particularly with the strikeforce they put out,” he said.
“Every game that’s how we’ve got to enter it. We might struggle to get seven, eight or nine wins we need, so we need to pick up points everywhere and that was a great point.”
Matt Phillips is close to a return but may still miss out with a hamstring injury along with Craig Dawson (knee) but Gareth Barry is fit again after a thigh problem.
Nacer Chadli (hip) and James Morrison (Achilles) remain out for Pardew.
He added: “We’ll make this clear, I’m going to give you no clues about what I’m going to do. One thing I know about Jose (Mourinho) is his mastery of the opposition is something that I want to get away from.
“I want to give him as few clues as I can, he’s the best manager in the world at subduing a team and therefore he will find it less difficult.”
Speaking to MUTV ahead of Sunday’s trip to West Brom, Mourinho offered an update on the Ivorian’s latest injury problem, which he had earlier described as “serious” without confirming an operation.
“He goes to surgery, the decision is made,” the Portuguese said.
“I don’t say (he’s out) for the rest of the season but it’s for the next two or three months.”
That length of lay-off puts Bailly’s hopes of taking part in February’s Champions League tie with Sevilla in doubt and also brings the January transfer market into view.
United are not short of numbers at the heart of defence, with Phil Jones, Chris Smalling, Marcos Rojo and Victor Lindelof all on hand, but Mourinho offered a mixed view of the mid-season market.
While insisting he does not approach the period with any great enthusiasm, he also made it clear that he could be tempted to do business if the right individuals become available.
“I’m not a big defender of the January market but there are players and situations that sometimes you have the chance to do and you shouldn’t lose that chance,” he said.
“I don’t like the January market just to buy, but if you can buy in January a player that you would buy in June or July and you have the chance to get him a few months before, then great.”
Speaking to #MUTV, Jose Mourinho has provided an update on Eric Bailly’s injury: “He goes to surgery, the decision is made. I don’t say [he’s out] for the rest of the season but it’s for the next two or three months.” pic.twitter.com/bsNF33o9So
— Manchester United (@ManUtd) December 15, 2017
One player who might fit that description further up the field is Antoine Griezmann, the France forward who has long been linked with the Red Devils.
His departure from Atletico Madrid appeared to move a step closer on Friday with manager Diego Simeone telling L’Equipe “of course he can leave at some point” and recognising the player’s need to “grow”.
But if Mourinho is eyeing an imminent move for one of European football’s most coveted talents, he is keeping his cards close to his chest.
“I don’t speak about players, especially players that belong to other clubs – there is no sense,” he said.
As for this weekend’s opponents, Mourinho offered a warm appraisal of a Baggies side who sit above the relegation zone on goal difference alone.
Tony Pulis is no longer around to spar with, having been dismissed and replaced by Alan Pardew, but Mourinho made it clear the Welshman’s legacy is still strong at the Hawthorns.
“They got so many points with Tony in the big matches,” he said.
“I think Tony assembled a very good squad, lots of players of experience, physical players adapted to the Premier League with know-how, lots of experience.
“So the squad is good, the players are good, the stadium is enthusiastic, the match is difficult.”
Manchester City's 4-0 drubbing of Swansea on Wednesday made them the first club ever in the top flight to win 15 matches in a row.
Starting with their first victory of this streak -- a 2-1 away success over Bournemouth -- up to their triumph at the Liberty Stadium midweek, the video above offers a closer look at how the Sky Blues went on this record-breaking run.