Verstappen, who had an error-strewn start to last season, eventually won two races - in Austria and Mexico - and finished only two points behind third-placed Kimi Raikkonen.
"I don't think he lacks anything to take those guys on and he's probably the driver they fear the most".
"For sure I'm going to be next to one of the best drivers on the grid at the moment and for me, it's a great challenge", Gasly told Motorsport.com.
But Horner expressed his confidence that the Dutchman, who became the sport's youngest victor when he won the 2016 Spanish Grand Prix at the age of 18, would be their heir apparent.
"His maturity has increased, we saw that during the course of past year", said Horner, though, other than the unnecessary clash with his teammate in Baku, many still question his actions in Brazil, not least the subsequent altercation with Esteban Ocon.
Verstappen hit the headlines after Brazil, when he was ordered to do two days of public service for shoving French driver Esteban Ocon after the collision.
"It was interesting talking to him after the Formula E race".
"I think his maturity has increased, we saw that during the course of past year. Whether you penalise or not.' I think that was actually a useful exercise for him to see the other side of the fence", said Horner.
Red Bull have taken a massive risk with Honda this season, a move that could work in both directions.
Now, the top team Red Bull is jumping aboard, and top official Dr Helmut Marko thinks Red Bull-Honda is on track for a world championship tilt.
But team boss Christian Horner says hopes are high that Honda has hit its targets and that it can at least make progress in 2019.
"Of course it depends on the strength of the competition, but on their own data, Honda has made some nice progress", Horner added. "[We have been in] very much a customer supplier relationship where effectively we were paying for a first class ticket and you get an economy seat".