PARIS—European cable magnate Patrick Drahi is trying to keep up his acquisition spree.

Altice SA, the Luxembourg-based telecommunications company controlled by Mr. Drahi, has offered about €10 billion, or roughly $11.4 billion, to buy French mobile operator Bouygues Telecom, according to people familiar with the matter, a deal that would increase Mr. Drahi's power in the French market by reducing the number of mobile operators from four to three.

As part of the offer, made about 10 days ago, Altice is proposing to merge Bouygues into Altice's existing French operations, called Numericable-SFR, one of the people said. That would make the combined group France's largest telecom company in terms of subscribers, overtaking former monopoly Orange SA.

Parent company Bouygues SA will consider the offer at a board meeting Tuesday, but it is unclear whether Bouygues Chairman Martin Bouygues, who controls the firm, will accept, the person added.

For Mr. Drahi, buying Bouygues Telecom would accelerate a $30 billion spate of purchases in the past year that have turned him into a player—albeit a highly leveraged one—on both sides of the Atlantic Ocean.

He most recently agreed to buy U.S. cable operator Suddenlink Communications for $9.1 billion. That followed an $8.4 billion deal last year to buy Portugal Telecom. Mr. Drahi has said that he plans to continue looking for deals in the U.S.

Mr. Drahi's biggest bets so far, however, have come in France. Just over a year ago, he beat out Mr. Bouygues in a takeover battle to buy Vivendi SA's SFR, which valued the company at €17 billion. He then swept in earlier this year to buy the 20% of the company Vivendi hadn't sold—faster than originally planned.

For Mr. Drahi, buying Bouygues Telecom would allow him to make more crucial cost savings in areas including network investments, marketing and information technology. Altice relies for the bulk of its revenue on Numericable-SFR.

At the same time, Mr. Drahi has also expanded in French media. Earlier this month, France's competition authority approved his deals to buy the newspaper Liberation and the magazine group L'Express.

Mr. Drahi's ambition to consolidate the fractious French telecom market could run into regulatory obstacles with the French government, which has recently opposed consolidation.

On Sunday, French Economy Minister Emmanuel Macron fired a warning shot against the deal, saying that companies should be focusing on investment and jobs, not mergers.

"The consequences of consolidation are negative in these respects, as several recent cases in Europe have proven," Mr. Macron said. "The time isn't right for opportunistic mergers that could benefit some but aren't in the public interest."

The comments come as the European Commission continues to take a tough stance on mergers in the telecom space, arguing that competition should be maintained to the benefit of the consumer. France's competition watchdog has also said it would be wary of such deals.

To address potential antitrust concerns, Mr. Drahi has struck a side deal with Iliad SA—owner of another mobile operator in France called Free Mobile—to purchase parts of Bouygues's telecom network and spectrum, the people said.

Since the arrival of Iliad's Free Mobile in early 2012 set off a price war, French telecom companies have said that their market is too competitive. Bouygues Telecom, the smallest of the three incumbent operators, has been one of the hardest hit by the price war. The company controlled by conglomerate Bouygues SA, whose operations also include construction and media businesses, last year slashed jobs to recover profits. Many analysts have said Bouygues Telecom will have trouble surviving as a standalone and some saw the job cuts as a preparation for a sale.

Mr. Drahi's Altice has been keen on a Bouygues purchase for some time, according to executives, because it would shrink the number of mobile carriers in France to three from four. But Bouygues has appeared unwilling to sell.

The price tag offered by Mr. Drahi is much higher than an offer for the company made by rival Iliad last year, which valued Bouygues Telecom at €5 billion to €6 billion, according to people familiar with the matter. Bouygues last year held talks with both Iliad and Orange to try to find a way to consolidate the market. But the three couldn't agree on terms and a price. Most analysts value Bouygues Telecom at around €5 billion.

French newspaper JDD first reported the news of Mr. Drahi's Bouygues bid on Sunday.

Write to Ruth Bender at Ruth.Bender@wsj.com and Sam Schechner at sam.schechner@wsj.com

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