Felipe Massa is expected to sign a one-year deal to remain with Ferrari next season, with a BBC report stating that he is going to be subsequently replaced by Vettel.
Rumours that the German has a pre-contract agreement with Ferrari have been ongoing for a number of months, and the fact that Ferrari allowed Sergio Perez to join McLaren from Sauber only saw the speculation continue to grow.
Red Bull Racing adviser Helmut Marko has previously revealed there are performance-related clauses that would allow Vettel to leave, despite his contract running until the end of the 2014 season.
But Di Montezemolo remains adamant that he wants a clear number one and number two driver line-up, and has denied that Vettel has signed an agreement with Ferrari.
"I've always said that I don't want two roosters in the hen-house," Montezemolo told Italy's Radio1. "I don't like that and it creates instability in the team."
Horner of Red Bull told BBC Sport: "Sebastian Vettel, without a shadow of doubt, will be part of the Red Bull Racing team in 2014."
Asked about the Massa situation at the Korean Grand Prix, Ferrari team boss Stefano Domenicali said: "Very soon you will know what will be our position on that respect.
"I think that for sure he understands now better the car, he understands better the tyres – and also the fact Felipe is a very, very strong driver because otherwise we would have already taken a different decision.
"So I feel that now when Felipe has under his control the understanding of the tyres and car, he performs at the level he should."
Vettel, meanwhile, has been installed as a strong favourite for the title following his win at Yeongnam on Sunday.
Horner said: "We've hit a purple patch of form at the right time but it's important we maintain that now. There's still a long way to go,"
Domenicali added: "Clearly at the moment Red Bull might seem unbeatable in everyone's eyes, but I can remember the same thing being said about McLaren on (the) Sunday afternoon in Singapore. The wheel turns quickly this year."