Ferrari's head of communications Maria Conti told reporters in a virtual press conference that the Italian team wanted to avoid the 'gossip' about Vasseur continuing over into this weekend's Bahrain Grand Prix in Malaysia.
"We have confidence -- full confidence -- in Fred Vasseur and in his team, and we continue to do so," she said.
The speculation was fuelled by comments made by former Red Bull principal Horner in an interview with the Times newspaper in which he said Ferrari were the only team he would consider joining without an ownership stake on offer.
Vasseur made clear his frustration at questions about the matter at last weekend's Azerbaijan Grand Prix in Baku.
"We would like to avoid that the next weekend in Malaysia will be focused only on this kind of gossip and speculation," said the spokeswoman.
Vasseur took over as Ferrari team boss in 2023 and was a key figure in bringing seven-time world champion Lewis Hamilton from Mercedes to Maranello last year.
Ferrari are second in the constructors' standings, 160 points behind Mercedes and 72 clear of champions McLaren. The Italian team last won an F1 title in 2008.