The CMA said Thursday that it would deepen its investigation after Microsoft informed the agency that it wasn’t offering concessions that could have gained it speedy regulatory approval.

Less than one-third of the more than 275 mergers the CMA has investigated since 2017 have moved to the second phase. Of those, the CMA blocked eight, more than a dozen were abandoned and others were approved or gained approval subject to certain conditions.

Microsoft referred to an earlier statement by President

Brad Smith

saying the company is ready to work with the U.K. regulator on next steps and to address any of its concerns.

Sony

didn’t immediately respond to a request for comment.

Microsoft’s deal for Activision is also being reviewed in the U.S. by the Federal Trade Commission. Since assuming office in 2021, FTC Chairwoman Lina Khan has taken steps to investigate more deals and what the commission sees as anticompetitive conduct.

Although the Microsoft-Activision deal is undergoing scrutiny in many markets, competition lawyers said regulators in the U.K., the European Union and the U.S. are the most likely to intervene. The EU hasn’t yet begun a formal review of the deal.

The U.K. could be looking to shelter its local companies, as it is home to some top game studios, including

Electronic Arts Inc.’s

Codemasters and

Take-Two Interactive Software Inc.’s

Rockstar North. It is the world’s sixth-largest videogame software market by revenue, according to industry tracker Newzoo BV. China is the largest, followed by the U.S.

Activision specializes in game development and owns some of the world’s most popular franchises such as “Call of Duty” and “World of Warcraft.”

Microsoft makes Xbox videogame consoles and its chief rival, Sony Group Corp., is known for its PlayStation system.

Nintendo Co.

has the Switch console, which can be played at home or on the go. All three companies also develop and publish their own games.

Microsoft is also one of the world’s largest cloud-computing service providers and has been a leader in the cloud-gaming market through its Game Pass subscription service. Sony offers cloud gaming through its PlayStation Plus subscription.

Competition lawyers said the positions of regulators around the world will depend in part on how they define the market affected by the Microsoft-Activision deal. If regulators define it broadly—for example, to cover videogames of all kinds—they may be less likely to view the merger as a threat to competition. A more narrowly defined market focused on videogames in the same shooter genre as “Call of Duty,” though, might produce more concerns, they said.

Another potential concern for regulators is whether the deal would give Microsoft an incentive to discriminate against Sony and Nintendo by restricting access to Activision games on those companies’ consoles, or by making it harder to play competing videogames on Xbox consoles, the lawyers said.

Microsoft has said that it will continue to make Activision games available for other consoles, but Sony said Microsoft’s pledge doesn’t go far enough because the company’s contract with Sony for keeping “Call of Duty” available on PlayStation consoles is good for only three years after their current agreement ends.

“We want to guarantee PlayStation gamers continue to have the highest quality “Call of Duty” experience, and Microsoft’s proposal undermines this principle,” Sony’s PlayStation boss, Jim Ryan, recently said in a statement.

Microsoft announced the agreement to buy Activision for approximately $95 a share in January. It has said it expects the deal, which would be its largest acquisition by far, to close by June of 2023.

Buying Activision would significantly increase Microsoft’s videogame revenue. Activision’s sales totaled $8.8 billion in 2021. Microsoft reported $16.23 billion in videogame revenue for the fiscal year through June, accounting for about 8% of its total revenue.

Write to Sarah E. Needleman at sarah.needleman@wsj.com and Kim Mackrael at kim.mackrael@wsj.com