Recently, the 19-year-old Dutchman and his father Jos have not categorically rejected rumours that Verstappen is linked with a future switch to Ferrari.
At the same time, even though an often clearly frustrated Verstappen is locked in a Red Bull contract until 2019, his car this year has been regularly breaking down.
When asked if he was relieved when Verstappen finally finished a race recently at Silverstone, Red Bull team boss Horner admitted: "You could say that."
Regarding Verstappen's recent outbursts, he told Algemeen Dagblad: "Of course he has shown some emotion, especially when you stick a microphone under his chin after a bad race, but he has also always been under control.
"This phase in his career is still character building for him, because F1 is not a sport that relies simply on a racquet, bat or ball. So he will only get stronger and more complete."
Horner said that Red Bull are at least making serious progress with pace and performance, after a less than impressive start to 2017.
"Since Barcelona, there has been real progress in every race with the possible exception of Silverstone, but we're clearly on the right track," he said.
Horner added that there is no doubt Verstappen will still be a Red Bull driver next year, but admitted that the energy drink-owned team need to make a step for 2018.
"It's up to us to give Max a car with which he can win before we talk about the future," said Horner.
"If we do that, I'm sure there's no team Max wants to race for other than ours."
Red Bull are third in the team standings heading into this weekend's Hungarian Grand Prix.