The F1 world champion has been strongly linked with a move to Mercedes and claims that Horner's departure from the team after 20 years will have no bearing on his decision.
Verstappen said ahead of the Belgian Grand Prix this weekend: "I don't think it will matter at all for my decision in the future.
"The only thing that matters is that we work on the car and make it as fast as we can make it, really."
The Dutch racer did hint that he could remain at Red Bull for next season and says his main goal is to be driving a quick car in order to compete for race victories and world titles.
Verstappen said: "The last one and a half years have not been what we want to be. Now we try to be more competitive this year a little bit, but for sure also with the new regulations."
Red Bull are languishing behind leaders McLaren this season but the four-time world champion has stressed that he is "very happy" at the team.
Quizzed on the prospect of not racing for Red Bull next year, Verstappen said: "There's also a possibility I don't wake up tomorrow. So then there is no driving at all. So, life is unpredictable.
"But in general, I'm very happy where I'm at - and I hope. That was still the target that was set out when we signed the new deal, you know, that I would drive until the end of my career."
Horner had been team principal at Red Bull since the team first entered F1 in 2005 but Verstappen says events like his sacking "can happen" in the world of elite sport.
He said: "Surprised. I mean, at the end of the day, in this world, things like that, they can happen.
"And when they told me, it's not like they just said, 'This is what we just decided', and then they hung up the phone. You know, you have a conversation about it.
"I don't need to go into detail what they said but it was, 'OK, if you guys think that this is the way forward, I'm the driver, you can decide, and this is how we're going to do it.'"