The Red Bull team boss has reacted to recent rumours suggesting that there is a performance clause in Verstappen's contract, allowing him to exit it early if an offer from Mercedes or Ferrari came his way.
Despite the reports, Horner insisted that the earliest Verstappen could leave the team would be 2021, adding that the talk of a performance clause was "purely speculation".
"We're very happy with him, he's happy within the team, I don't see there being any changes for next year," Horner told Sky Sports F1.
"It's more about 2021, when all of them are out of contract, than for 2020. I heard this thing about 'he has to win a race'. That's not the case at all. There's always speculation at this time of year.
"The contract between the driver and team will always stay confidential but it's safe to say that every driver in this pit lane has got clauses in their contract.
"There's been speculation about Max's contract and it's purely speculation. He's very happy within the team, he believes in this project and days like today only endorse that. I have no doubts about Max."