Ms. Enders and other analysts believe that the deal will ultimately be approved, but it is also clear that Sky is an asset the Murdochs do not want to risk losing out on a second time. That appears to be particularly true of James Murdoch, who is the chief executive of 21st Century Fox and the chairman of Sky.

“It will set up his future as a chief executive officer,” Ms. Enders said.

Sky is both a giant cable and broadband provider in Britain and a producer of shows like “The Young Pope,” the Jude Law vehicle that Sky developed with HBO and Canal+. It has divisions in Italy and Germany and deals with Britain’s Premier League and the Bundesliga soccer league in Germany. Its Sky Atlantic channel has rights to American hits like “Game of Thrones.”

From a business perspective, the timing of the deal is good.

Sky has a reasonable valuation, and it has become highly profitable. It also has direct experience with consumers that 21st Century Fox lacks, as a provider of cable and broadband services and through NowTV, its rival to Netflix. That expertise is valuable at a time when many consumers are abandoning bundled cable offerings.

“A generic problem for a lot of media companies everywhere, in the U.K. or the U.S., is, how do you crack the direct-to-consumer nut?” said Thomas Singlehurst, an analyst at Citigroup. “And here is a company that has been doing it very successfully, and that’s part of the consideration alongside the cheap valuation, the weak sterling, and the fact that the group just throws off a lot of cash.”

Mr. Singlehurst said he expected that the government would ultimately approve the deal.

“At the end of the day, you do have to have proper evidence of something that has gone wrong, and that evidence has to have been provided definitively one way or the other, most likely in a court of law,” he said.

With the fit-and-proper test, Ofcom is trying to determine whether the people who would be running the company are fit to do so. While relatively rare, the test is familiar to the Murdochs. The last such review by Ofcom began in 2011, and also examined the fitness of Sky, then called British Sky Broadcasting, amid a bid by News Corporation to acquire full control of the company.

News Corporation has since split into two companies, including 21st Century Fox. Sky ultimately passed the review, but James Murdoch was criticized in Ofcom’s report for mishandling the hacking matter, and the proposed acquisition was abandoned.