While Mercedes have dominated the season with six wins from six races, Horner and his four times world champion Sebastian Vettel were not giving up hope after arriving in Montreal for Sunday's Canadian Grand Prix.
"Nobody is chucking in the towel," said Horner, even if the standings suggest the title has become a duel between Mercedes team mates Lewis Hamilton and Nico Rosberg.
While Rosberg leads overall with 122 points to Hamilton's 118, Australian Daniel Ricciardo has only 54 as the closest Red Bull driver.
"We have a mountain to climb; Take nothing away from Mercedes, they have done an incredible job this year but as we've seen in previous years, we had a difficult start in 2012 and managed to turn that around," said Horner. "Nobody is giving up on anything."
Mercedes have a hefty 240 points in the constructors' standings, with Red Bull second on 99.
The champions have been playing catch-up with an under powered Renault engine but there have been signs of progress. Ricciardo made it onto the podium at the last two races with third place finishes in Spain and Monaco.
The Australian is fourth overall, with Germany's Vettel sixth.
It is an unfamiliar position for a team that has won both titles for the past four years but Horner underlined his faith in Renault, denying rumours that the team was considering its engine options.
"Renault fully understands the predicament they are in," said the Briton. "There are no ultimatums for Renault other than we want to improve as quickly as we can.
"We have enjoyed a great deal of success with them over the years, all our world champions and victories have come with Renault engines and we trust in them to sort their issues out."
(Editing by Alan Baldwin)
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