The offer values Sky plc, the owner of Sky News, at £26bn.
Comcast said its latest offer – up from the £12.50 per share it offered in April – had been recommended by Sky’s independent committee of directors.
Brian Roberts, chief executive of Comcast, said: “We have long admired Sky, which we believe is an outstanding company and a great fit with Comcast.
“Today’s announcement further underscores this belief and our commitment to owning Sky.”
Fox’s latest £14 per share bid, announced early on Wednesday, was a substantial increase on the £10.75 it first put forward in December 2016, and also trumped Comcast’s initial £12.50 offer.
Fox is already Sky’s biggest shareholder, with a 39.1% stake.
Its bid to buy the rest of the shares has been delayed by a series of regulatory investigations, first by Ofcom, the telecoms and broadcasting regulator, and then by the Competition & Markets Authority, the UK’s competition regulator.
It advised the government in May that the deal should be allowed to go ahead but only if it sold Sky News to avoid concerns over plurality of media ownership in the UK.
Fox’s biggest shareholder is the Murdoch Family Trust, which is also the biggest shareholder in News Corporation, owner of The Sun, The Times and The Sunday Times, some of Britain’s bestselling and most influential newspapers.
The government is expected to give its blessing to the Fox offer, subject to those conditions being met, later this week.
Announcing its new offer on Wednesday, Fox said: “As the founding shareholder in Sky, we have remained deeply committed to bringing these two organisations together to create a world class business positioned to deliver the very best entertainment experiences well into the future.
“We strongly believe that a combined 21CF and Sky will be a powerful driver for the continued growth and vibrancy of the UK and broader creative industries.
“The enhanced scale and capabilities of the combination will enrich Sky’s ability to continue on its mission for years to come, especially at a time of dynamic change in our industry.”
Fox agreed in a separate deal just before Christmas to sell its entertainment assets, including Sky, to Walt Disney for $65bn in shares.
Comcast subsequently made a rival offer of $66bn in cash for those assets – prompting Disney to sweeten its terms last month to $85bn, including debt, in cash and shares.