The seat alongside Verstappen has become the source of great scrutiny, with Albon promoted to replace Gasly at the Belgian GP after the Frenchman's under-performance so far this season.
With two drivers previously dropped from the senior team and another in his rookie season, it has been suggested that Red Bull could benefit from a more experienced head, such as Hulkenberg, who is on the market after Renault confirmed he will be replaced by Esteban Ocon next year.
Speaking of the decision to demote Gasly, Horner says the Frenchman has the chance to "rebuild" his reputation in the junior outfit.
"We took a little time to reflect after Hungary," Horner told Sky Sports.
"Hungary was a tough weekend for Pierre and we came to the conclusion that it was almost fairer for Pierre to say, look, we have four cockpits in Formula 1, let's take the pressure off him as he was obviously struggling with that pressure.
"The situation was getting worse and worse and now he has the time at Toro Rosso to rebuild himself. We now give Alex a chance as we know Daniil Kvyat very well and we see his potential. With Alex, we see his potential too.
Horner confirmed that the plan is to pick a driver from those within the Red Bull stable, but admitted the likes of Hulkenberg could yet stand a chance.
"That's the pool we are looking at," Horner said of Albon, Gasly and Kvyat.
"It does depend on how things pan out over the next few races but ideally we're going to be looking at within that pool of three.
"Obviously we monitor the situation externally but it is unlikely that we would take a driver from outside our own development program.
"But as I said, if we don't feel we have a driver that can get the job done, as we did with Mark Webber, he was taken externally."
GPFans is a multi-platform, multi-language brand dedicated to Formula One coverage. We bring you all the ins and outs of the sport, 24/7, everything from up-to-the-minute news and features to the latest viral stories and clips.
We believe that a new generation of exciting, outspoken drivers will make F1 more popular than ever before, and we want to give our users access to as much of their heroes as possible, on and off the track. From Lewis Hamilton to Max Verstappen, Daniel Ricciardo to Sebastian Vettel, we provide in-depth analysis of every every Grand Prix in the season, from Australia to Abu Dhabi.
With Formula One under the new ownership of Liberty Media, how the sport is being covered is evolving, and GPFans will look to be at the heart of this progression into new media, as one of the fastest-growing sites covering the king of motorsports.