Out-of-contract at the end of the season, Ricciardo has been the subject of speculation for weeks as he weighs up his options for 2019, though Horner suggests it's now certain he'll remain with Red Bull.
The Australian has been linked with moves to Mercedes, Ferrari, Renault, and even McLaren.
“We've now got to the point where there is a desire from both sides to continue the relationship,” said Horner.
“So it's just a matter of now really going through the detail and coming to a conclusion which will certainly be done prior to the summer (August) break.”
Ricciardo has made no secret of his desire to have a championship winning car at his disposal, with Mercedes and Ferrari currently in the best position to provide that.
However, according to Horner, no offer was received from either team.
“I think that had Mercedes put a serious offer on the table, or an offer, he would have been potentially interested,” he claimed.
“He's impatient to try and challenge for a world title so I think he would have had his head turned in the event of a Mercedes or a Ferrari offer but I think those two teams look pretty set where they are.”
Hamilton is expected to be confirmed in the coming days, while the unlucky Bottas also seems set to be retained for a third season with the team.
At Ferrari, Sebastian Vettel holds a contract with the Scuderia for next year.
It makes Ricciardo's only front running option Red Bull, which is set to switch to Honda power for next season.
“Obviously, one of the things he was waiting to understand was where we were going with engines,” Horner reasoned.
“He's seen and understood the decision making behind where we've gone.
“Hopefully, the rest of it becomes a formality, unless Lewis announces his retirement at the British Grand Prix this weekend.”
The British Grand Prix begins on Friday with opening practice at 1900 AEST.