Mekies has agreed to leave Ferrari and join AlphaTauri as its new team principal to replace Franz Tost, who is stepping down at the end of the year.
But no date has been set for Mekies' arrival as he remains under a long-term contract with Ferrari, so his release needs to be agreed.
Tost himself said the matter would likely be one for Red Bull and Ferrari to resolve.
"First of all, it's between Laurent and Fred [Vasseur]," said Tost about the process of sorting Mekies' arrival.
"Then, of course, Red Bull will take a major role in this. I think that Oliver Mintzlaff [Red Bull CEO] will then talk to the CEO from Ferrari and find a solution."
Sources have suggested that Ferrari could be willing to let Mekies go early if it can get hold of some Red Bull personnel early that it has signed but are currently on gardening leave.
It comes amid rumours that Ferrari has also been trying to poach more senior technical staff from its Milton Keynes-based rival.
Speaking at the Miami Grand Prix, Horner made it clear that senior Red Bull staff who had been part of the Ferrari rumours – including technical director Pierre Wache – had emphasised their total commitment to his team.
And, while he did not dismiss outright the prospect of an agreement between Red Bull and Ferrari over Mekies, he said it would not revolve around top-level personnel.
"Will there be a hostage exchange for Laurent Mekies?" said Horner. "Well, we don't have any hostages.
"In terms of personnel moving to Ferrari, certainly at a senior or medium level, there's nothing planned.
"Individuals that have been mentioned in connection with Ferrari came to see me last week and mentioned their disbelief in some of these rumours.
"But, you know, it's Formula 1, and that will inevitably happen. But there's no plans for any senior members of our team to be joining Maranello."
As Motorsport.com revealed earlier this week, Red Bull has recently been boosted by its chief technical officer Adrian Newey agreeing to extend his contract at the team.
