The proposed offer is worth £10.75 a share in cash, a premium of 36% to the closing price on 8 December.
Sky shares ended 26.6% higher at 999.8p in London following the announcement.
Media tycoon Rupert Murdoch controls 21st Century Fox, which already owns a 39.1% stake in Sky.
Sky said that the independent directors of both companies had "reached agreement on an offer price" of £10.75 a share, but added that "certain material offer terms remain under discussion".
Fox said Sky's directors had indicated that they were willing to recommend the proposed offer.
According to Reuters' calculations, Fox would pay £11.25bn for the stake in Sky that it did not already own.
Sterling's 16% fall against the US dollar in the wake of the Brexit vote has made UK companies more attractive targets for foreign companies.
Fox is required to "clarify its intentions" by 6 January, or walk away for at least six months under UK takeover rules.
Alex DeGroote, analyst at Peel Hunt, said it was "not quite a done deal", but he would be surprised if it did not go ahead.
"Sky has not performed well in the UK stock market this year, and is seen as a Brexit loser. Fox is of course also a dollar bidder, and the collapse in sterling makes Sky a less expensive purchase than pre-Brexit," he said.
"There will also be cost synergies, which will reflect economies of scale in technology and content, such as sports and movie rights."
Fox chief executive James Murdoch was named chairman of Sky this year, fuelling speculation that the US media company would make a bid.
Almost 30% of Sky shareholders voted against the appointment of James Murdoch as its chairman at the annual meeting in October, with some saying he was too closely linked to Fox.
Piers Hillier, chief investment officer of Royal London, which owns a 0.35% stake in Sky, said at the time: "Should Fox make a bid for Sky, investors need a strong independent chairman to protect the interests of minority shareholders and negotiate the best possible deal."
Rupert Murdoch has sought to take full control of Sky for many years.
In June 2010, his company, News Corporation - from which 21st Century Fox was subsequently split off - made a 700p-a-share offer that valued Sky at about £12bn. It was rejected by Sky's directors for undervaluing the company.
The bid was ultimately abandoned in mid-2011 in the wake of widespread opposition and the fallout from the phone hacking scandal that prompted the closure of the News of the World newspaper.
Vince Cable, the Liberal Democrat business secretary at the time of the 2010 bid, said the new offer threatened media plurality in the UK.
"The way Theresa May's government deals with this is a test of their independence from the influence of large proprietors."
Deputy Labour leader Tom Watson said it was up to regulators to ensure competition concerns were addressed.
"The bid must also be judged on its likely impact on the UK news market and the provision of robust and independent journalism," he said.
"Finally, given the likely concentration of further media power in the hands of a single company, it is right that the 'fit and proper' test should be applied by Ofcom if the deal is approved by Sky shareholders."
News Corp owns newspapers including the Sun, the Times and the Wall Street Journal, as well as other assets such as publisher Harper Collins.
In July 2014 BSkyB, as it was then known, paid almost £5bn to take over Rupert Murdoch's pay TV companies in Germany and Italy.
US media giant 21st Century Fox has made a takeover approach for Sky that values the UK-based satellite broadcaster at £18.5bn.