The ex-Renault F1 boss was seen having a "long conversation” with Red Bull team principal Horner on the terrace of the team’s hospitality suite during the Qatar weekend, according to The Daily Mail.
But Red Bull insiders reportedly responded “not a chance” when a suggestion was put to them that the meeting might have been about signing Alonso to replace the under-pressure Sergio Perez next season.
Perez’s position at Red Bull has come under enhanced scrutiny in recent weeks amid his ongoing torrid run of form that has seen him pick up just five points from the past three races.
The Mexican has struggled to match the performances of his Red Bull teammate Max Verstappen, who has won 14 of 17 grands prix on his romp to clinching a third consecutive world title.
While Verstappen has notched up a remarkable 433 points this year, Perez has managed just 224, leaving him only 30 points clear of Mercedes’ Lewis Hamilton in the battle for second place in the constructors’ championship.
There has been speculation that Perez, who holds a contract for 2024, has been told by Red Bull that he must secure runner-up spot in the championship, or he will lose his seat.
Red Bull have denied giving Perez an ultimatum.
Perez, meanwhile, laughed off what he described as false reports claiming he is set to retire from F1 at the end of the season.
"That really sums up my season: a guy says something about me and all of a sudden it becomes true," Perez told reporters ahead of the United States Grand Prix.
"But I have a contract for next year and I have no reason to not fulfil that contract.
"I'm going to give my very best to it, I've done a commitment but it will not be my final contract in F1."
Asked if he is certain he holds a watertight contract for 2024, Perez replied: "100 percent.
"I have a contract and conversations with the team as well. There is no reason for me not to fulfil that contract."
While the prospect of an Alonso-Verstappen driver line-up at Red Bull would certainly set tongues wagging, it appears unlikely for the time being.