Reports on Monday suggested the 25-year-old Vettel, the two-time defending world champion, had been lined up to switch to Ferrari in 2014, with Felipe Massa retaining his seat next season as a stop-gap.
That would see Vettel become the team-mate of his chief rival for the 2012 world title, Fernando Alonso, who is contracted with the Maranello team until 2016, but Montezemolo argued that would go against the team's long-standing policy of having a clear number one driver on their team.
"Today [Tuesday] the problem is not with drivers," he said on the team's official website. "2014 is still to come but I don't want to have two roosters in the same henhouse, rather two drivers who race for Ferrari and not for themselves.
"I don't want problems and rivalries, which we didn't have between (Michael) Schumacher and (Eddie) Irvine, between Schumacher and (Rubens) Barrichello, between Alonso and Massa or Massa and Schumi or Massa and (Kimi) Raikkonen."
Red Bull have also rejected the report, with team advisor Helmut Marko telling German newspaper Bild: "These are untruths. This report is simply not right."
Team boss Christian Horner told BBC Sport: "Sebastian Vettel, without a shadow of doubt, will be part of the Red Bull Racing team in 2014."
Vettel moved ahead of Alonso in the 2012 driver's championship standings following yesterday's [Sunday] Korean Grand Prix, with his victory and Alonso's third place translating into a six-point advantage.
Alonso clashed with Lewis Hamilton when the two were allowed to compete as rivals while team-mates at McLaren, and is said to have a clause in his contract which gives him approval over the identity of his Ferrari team-mate.
There has been speculation over the future of Massa with Ferrari all season after a series of indifferent performances, but he has improved of late and took fourth place yesterday [Sunday].
Montezemolo sounded positive about the Brazilian's future.
"We have the best team: yesterday too, the work on track was perfect," he said. "We also have the best driver in Fernando Alonso and I'm happy to see Felipe Massa back at a high level: tomorrow [Wednesday] I will meet him at Maranello, we will have a chance to talk about his future and then I will take a decision."
He added: "You can see from the constructors' championship that he has returned to the levels that we used to know well. I expect a lot from him and his capacity to take points from our rivals."
Today's [Tuesday] speculation also suggested respected designer Adrian Newey could make the move from Red Bull to Ferrari, but this was likewise rejected by Montezemolo.
"As for Newey, I can say that he is very good at his job but we also managed to win eight constructors' titles in the last 13 championships without him: I have huge faith in our own engineers," he said.