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The Wales midfielder had himself warned that he was a bit short of fitness after his surprise loan move from Juventus on Monday.
Asked by Sky Sports News ahead of Wednesday's visit to Celtic Park if Ramsey could feature, Rangers boss van Bronckhorst said: "No, Aaron will not be in the squad. He's been out for a couple of weeks, he trained with the squad today, he looks really good, but tomorrow comes too soon for him.
"We'll assess him this week but I think we can welcome him really quickly to the squad."
Ramsey could have a part to play when Rangers host Hearts at the weekend.
Asked if Ramsey's injury issues had been a worry in bringing the midfielder to the club, Van Bronckhorst said: "No, we had a good discussion. We spoke with Aaron for the first time at the weekend, talked about his situation, talked about his physical ability at the moment.
"For me, it was clear that we wanted to sign him because I think he will be a great asset to the team. And of course now with the last months of the season on the way, he can help us be successful."
Ramsey's arrival was the highlight of a busy Deadline Day for Rangers, and Van Bronckhorst is thrilled with the club's business in the January transfer window.
"Hard work of course from [sporting director] Ross Wilson and the club [was so important] to be able to bring in a player with the quality of Aaron," the Dutchman said.
"We were really pleased we could sign him yesterday. It was a busy day for us, we also signed Mateusz (Zokowski) from Poland, a full-back on the right, so I'm more than happy with the last day.
"I'm pretty pleased also to welcome Amad Diallo from Man Utd, and we saw at the weekend what kind of quality player he is. I think it is always important when you go into a window that you leave that transfer window stronger and more competitive and I think this window we have achieved that."
Celtic were beaten by Rangers earlier this season and trail the champions by two points ahead of Wednesday night's derby.
Hoops boss Ange Postecoglou believes Ramsey's arrival at Ibrox is good for Scottish football but cautioned that performances are more important than profile.
When asked about the signing of Ramsey, Postecoglou said: "He's a quality player and I still think he is a very good player when you look at the level he has been playing at for the last couple of years.
"Club wise he hasn't played a lot, but when he played for the national team in the summer he still looked a very, very good player.
"So it's good for the league, but ultimately with all these signings, whether they are high profile or not, it comes down to performances.
"No-one had heard of Kyogo before I brought him here but no-one's going to tell me he's not good for the Scottish game."
Can Ramsey still cut it?
Sky Sports' Jack Wilkinson:
Ramsey was widely applauded for taking the leap to go and play overseas, and it's a decision that has almost certainly been vindicated.
Despite not quite hitting the levels he did in his final season at Arsenal, Ramsey's first two campaigns at Juventus under Maurizio Sarri and then Andrea Pirlo were a success and saw the Welshman win Serie A, Coppa Italia and Italian Super Cup titles.
Issues have cropped up this season under Massimiliano Allegri, however, mainly in the form of injuries, which have limited him to just three Serie A appearances this season.
In fact, Ramsey has missed 17 games this season, as many as he did in his first two seasons combined at Juventus.
Ramsey will undoubtedly look back at his time at Juventus, the success he's enjoyed, and the opportunity to experience a new footballing culture, fondly, but for a player aged 31, and with a decent chunk of his career remaining, he cannot be appearing as infrequently as he has.
With Juventus reportedly keen to get Ramsey's hefty salary off the wage bill, moving on from the club could be one that suits both club and player.
A clearer indication of the prowess Ramsey still possesses, and his possible suitability to Rangers, can be seen upon studying his performances for Wales since he moved to Juventus, rather than his final-season struggles in Turin.
Remarkably, Ramsey has scored more goals for his country, 13, since moving to Juventus than he actually has for Juventus, and that's despite playing three times the number of Serie A games.
In terms of his name and reputation alone, three-time FA Cup winner Ramsey would definitely sprinkle some stardust on Giovani van Bronckhorst's squad, who head into 2022 in position to retain the Scottish Premiership title.