Webber was left infuriated by team-mate Sebastian Vettel's actions in the Malaysian Grand Prix at the weekend after the German ignored team orders to pass Webber and win the race.
In the aftermath of the controversial manoeuvre, Webber said he would be returning to his home in Australia to think over his future.
However, Horner says he is convinced that Webber will remain a valued member of the reigning constructors champions.
"I've got no doubt that Mark will see out the season with us," Horner told Sky Sports News.
"Mark knows there was no conspiracy within the team," he added.
"Our intention was to shut the race down, to minimise the risk ... the intent of the team was for Mark to win that race. We didn't suddenly give Sebastian an instruction to say, 'go and pass your team-mate'.
"He is big enough to know there was no malice, there was no intent from the team to create any situation like that.
"He is in a car that is capable of winning Grand Prix and hopefully going for a championship."
Red Bull hinted that he may even stay for 2014.
"It's pure speculation that Mark will not drive with the team in 2014," the team stated when asked about comments by Webber's manager Flavio Briatore that he was sure the current line-up would not continue.
"Mark's contract has been renewed for the last few years on an annual basis - he has continually chosen to drive for the team each year and the team has similarly wanted to continue working with him as he is an excellent driver and a competitive racer.
"We are two races into the current season and it's far too early to be talking about 2014."
Webber, at 36 the oldest driver in Formula One, had turned his engine down in response to the orders from the pitwall.
Others, including Briatore, questioned who was really in charge at Red Bull.
The Italian, who was banned from the sport after the Renault team he led were found to have ordered Brazilian Nelson Piquet Junior to crash deliberately to help his team-mate Fernando Alonso, accused Horner of weakness in a RAI radio chat show in Italy.
Red Bull pointed out that a weak principal would not have steered the team to three successive titles - a feat only ever managed by three other outfits - while managing such a competitive pairing.
The team added that the pairing of Webber and Vettel had won 35 races, taken 80 podiums and 13 one-two finishes since their first season together in 2009.
"This successful period includes some spells of intense on-track rivalry between the two drivers, which began in Turkey 2010 and has seen both drivers ignoring team orders at different times," it said.
"The team has managed the situation each time in its own way behind closed doors."
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