Sky told the Competition and Markets Authority (CMA) that the regulator should not assume "the continued provision" of Sky News.
BBC media editor Amol Rajan said it was a "credible threat".
Fox owns 39% of Sky but wants full control of the satellite broadcaster.
In a submission made to the CMA last month, but published by the regulator on Tuesday, Sky said it "would likely be prompted to review" its position if "the continued provision of Sky News in its current form unduly impeded merger and/or other corporate opportunities available in relation to Sky's broader business".
This would particularly be the case if shareholders objected to the merger not happening, Sky said.
Closing Sky News would only be an option of last resort, and the broadcaster would try to find a buyer for the media company before that eventuality, the BBC understands.
'Alarming'
"The messaging coming through is alarming for supporters of Sky News but it runs completely counter to all the investment that there has been in the channel in all the recent months and years," said Joey Jones, a political correspondent at Sky News for 16 years and now head of public affairs at PR firm Weber Shandwick.
But he said the threat was a risky move by Sky: "Inevitably this will be perceived by those who are already hostile to the proposed takeover, particularly in the political arena, as sabre rattling and as a perceived threat by the company".
Losses
Media editor Mr Rajan said that Sky News lost "an awful lot of money".
"It loses tens of millions of pounds, and I think the independent directors of Sky are sending a very clear message... that if they had to choose, maybe they'd prefer for commercial reasons to do the deal with 21st Century Fox rather than continue to fund the losses at Sky News."
The submission comes a day after reports that Fox has discussed selling "most" of its business, including its Sky stake, to Disney.
Fox has faced a number of hurdles in its bid to buy Sky, including the CMA investigation and opposition from some politicians.
Some fear the deal would give Rupert Murdoch's family too much control over the UK media.
The Murdoch family owns controlling stakes in both News Corporation, which owns UK newspapers such as the Sun and the Times, as well as Fox, which operates in film and TV.
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