Lead driver Max Verstappen is tipped by Horner to rattle his rivals this year after his strong end to 2018, recording five podiums in the final five races.
"Max, if you look at his performance in the second half of the [last] year, was the second highest points scorer to Lewis", Horner said.
"In the next 12 months, I still think, and everyone would say this, that Mercedes and Ferrari are still the teams to beat. I think he's probably the driver that they fear the most".
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.
Former champions Red Bull, who finished third overall behind champions Mercedes and Ferrari past year, have switched from Renault power units to Honda for the season starting in Australia on March 17. I don't think we will match Ferrari and Mercedes straight away in terms of power, but Honda is giving everything to get there as quickly as possible. He is worldlier, more experienced. "He's just in a better place to be able to deal with the pressures that are placed on him".
He said Verstappen had learned from his mistakes and that the 'community service' charge he served for starting an altercation with Force India's Esteban Ocon following their crash in Brazil a year ago had opened his eyes. Not that he was particularly impressed by Formula E, but he came way from being with the stewards and said: "'Crikey, these guys have got hard decisions to make in terms of do they impose a penalty or do they not'".
"Progress is our target", Horner said, as quoted by BBC Sport. Horner says the aim is to be competitive on a wider range of tracks this year, and he is happy with the progress Red Bull has seen from the Japanese manufacturer during the off-season.
Horner said Red Bull aimed to be more consistent across all circuits this year, rather than just those where engine power was less of a factor.
Honda supplied Red Bull's junior team Toro Rosso past year.
"It is a new start with Honda, everyone is very motivated", Verstappen told his official website.
"An very bad lot of frustration was born out of that.so I think with Honda, it being a true technical partnership, there's much more collective responsibility from both sides rather than being a customer-supplier scenario".