Fox, which now owns just 39 per cent of Sky, expressed confidence either option would address the CMA's outstanding concerns.
Fox is offering to sell Sky News channel to Disney to ease its acquisition of British pay-TV platform Sky.
United Kingdom regulators have expressed concerns that if Fox acquires Sky News, Fox chairman Rupert Murdoch will own several of the country's largest media outlets.
Critics have voiced concern that the deal would give the Murdoch family, which controls Fox, too much influence over Sky News. And it is competing against Comcast, the USA cable giant that plans to make its own bid for control of Sky.
"21st Century Fox (21CF) proposed new ring-fencing remedies which would effect the legal separation between Sky News and the rest of Sky, establishing Sky News as a distinct company within the Sky Group, with its own fully independent board and under the management control of the head of Sky News", it said.
Fox said its new concessions went beyond the steps that Britain's media regulator Ofcom said would mitigate concerns about Murdoch's influence.
Since unveiling the 21st Century Fox bid in 2016 for the 61 percent of Sky that it did not already own, Murdoch has faced sharp questions over whether he would oversee too much of the media in Britain. Disney also said it would "agree to sustain the operating capital of Sky News and maintain its editorial independence".
The group, which has a broadband arm serving 29 million USA customers, owns NBC and Universal Pictures and wants to seize more than 50 per cent of Sky.
21st Century Fox anticipates that it will acquire the remainder of Sky by 30 June. Funding for Sky News would also be guaranteed for 15 years.
Sky now trades higher than the Comcast proposal, indicating that investors are expecting a bidding war.
He went on to say that Sky's comment telling shareholders to "take no action at this stage" was: "somewhat of a coded signal to Sky shareholders not to accept the Comcast bid as there may be a counter-offer coming through from Fox (and, by implication, Disney)".
"We have worked diligently with the CMA throughout its extensive review".
Even if Fox's proposals satisfy Britain's government and competition regulator, however, it may still need to raise its recommended offer for Sky after USA cable group Comcast Corp said it meant to make a higher counter-bid.