Rumours had swirled around the paddock at the Singapore Grand Prix on Friday after two European publications, Auto Motor und Sport in Germany and Italy's Gazzetta dello Sport, claimed Red Bull and Aston Martin had broken the 2021 spending ceiling.
The governing body FIA introduced the budget cap, set at $145 million, for the first time last year to make racing more competitive.
Ferrari racing director Laurent Mekies told the BBC: "It's now no secret that two teams broke the 2021 budget cap."
And Mercedes team principal Toto Wolff told Sky Sports F1 that the breaches were "an open secret in the paddock".
But a tetchy Horner hit back Saturday as he told reporters in Singapore he wanted the accusations withdrawn.
"We were a little bit taken aback by comments that were coming in from two of our rival teams yesterday," Horner said.
"The submission to the FIA is confidential. I would be intrigued to know where their source of information for these fictitious claims have come from.
"They're hugely defamatory. We take umbrage to them. Unless there is a clear withdrawal of those statements, we will be taking it incredibly seriously and looking at what the options available to us are."
The 10 teams had to submit their audited accounts to the sport's governing body FIA by March 31, who are investigating if anyone has broken the budgetary rules.
"We made our submission in March and we stand by that 100 percent behind that submission that we are below the cap," said Horner.
"For teams, that are not party to any form of the submission, to be making claims about our submission and prospective penalties is just bang out of order."
- 'Unsubstantiated' -
The FIA is due to report its findings next week and late Friday issued a statement, calling the paddock rumours "unsubstantiated".
"The FIA is currently finalising the assessment of the 2021 financial data submitted by all Formula 1 teams," it said
"Alleged breaches of the Financial Regulations, if any, will be dealt with according to the formal process set out in the regulations.
"The FIA notes significant and unsubstantiated speculation and conjecture in relation to this matter and reiterates the assessment is ongoing and due process will be followed without consideration to any external discussion."
Horner could not hide his fury at what had unfolded on the first race weekend where Red Bull's Max Verstappen has a mathematical chance to clinch this year's world championship.
"It is absolutely unacceptable to be making comments of the type that were made yesterday," said Horner.
"One can only assume it is not a coincidence, you know, that this is at the point where Max has his first shot at winning the world title.
"And here we are talking about nothing but cost caps rather than the phenomenal performance that he has had this year."
The FIA views any overspend of less than five percent of the cap to be "minor" with penalties ranging from fines to points deductions for drivers and constructors.
A more serious "material" breach carries a maximum penalty of expulsion from the championship.
If Red Bull are found to have breached the cap, it would cast a further shadow over Verstappen's first world title, won in controversial circumstances in the final race of 2021 in Abu Dhabi.
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