Speaking to the Daily Mail ahead of the 2023 Abu Dhabi Grand Prix season finale, Red Bull team principal Horner claimed the Hamilton camp had sought out a potential seat at the reigning constructors' champion.
He said that out of several approaches over the years, "most recently, earlier in the year, there was an inquiry about whether there would be any interest".
But Hamilton has rebuffed Horner's version of events, instead claiming that the Milton Keynes team boss messaged the Mercedes driver first.
Speaking on Thursday in Abu Dhabi, Hamilton said: "No, I didn't approach them… Christian messaged me.
"I've checked with everyone in my team, and no one has spoken to them. But they have tried to reach out to us."
Hamilton reckoned Horner had texted his old phone number, and he only saw the message months later but did then reply.
According to his version of events: "I picked up my old phone, which I just found at home, that had my old number on it.
"I switched it on and obviously hundreds of messages came through and I realised there was one from Christian to get together and have a chat at the end of the season…
"I just replied to him on my new phone. It was after a weekend. It was quite late on that I found the message. It was from earlier on in the year. So, it was like months later."
Hamilton says Horner was not specific about why the pair should meet, but he replied to congratulate Red Bull on its dominant year, adding that he hoped Mercedes could bring the fight in 2024. Horner apparently returned the sentiment.
"There's a lot of people here that like to drop my name in many conversations because they know it's going to make waves," Hamilton speculated.
"If you're a little bit lonely and aren't getting much attention, it's a perfect thing to do. Just mention my name."
In late August, Hamilton signed a new two-year contract extension to remain at the team with which he has won eight constructors' championships since joining from McLaren for 2013.
Despite both Hamilton and team boss Toto Wolff saying early on that negotiations would be swift, final confirmation of the new deal was delayed.
Hamilton continued: "I did tell Toto [about replying to Horner], especially when the story broke.
"Also, I wanted my team to know because people think those things and it's never a positive."
Hamilton reckoned any driver would want to drive the Red Bull, calling the RB19 the "most dominant car of all time".
But he believed there was more satisfaction to be had in guiding Mercedes back to the top after two seasons of faltering under the new ground-effects rules. This, he added, would be better for his "legacy".
Horner also suggested that Hamilton joining his team would not work out due to the internal relationship with bitter rival Max Verstappen. But Hamilton reckoned this would not be an issue.
He said: "I'd be more than happy to race against Max in the same car. That would be wonderful. I don't think he wants me to be his team-mate."
