PARIS--Vivendi SA (VIV.FR) is pushing ahead with roughly EUR1 billion in cost cuts at its French telecoms unit, just days after Vivendi's chief executive walked away amid disagreement over whether the entertainment and media conglomerate should split itself apart.

The French telecom unit, dubbed SFR, said in a meeting with union officials on Tuesday morning that it plans to cut roughly EUR450 million in costs in 2012, and roughly an additional EUR500 million in 2013, according to people familiar with the matter. The cost-cutting figures came as part of a broader strategic plan to counter a price war in the French phone market.

SFR also told unions it will announce in November a round of job cuts through voluntary departures, a spokesman said. The spokesman declined to specify the number of cuts, saying details of SFR's broader reorganization were not yet decided. SFR has about 10,000 employees.

The cost-cutting comes just days after Vivendi chief executive Jean-Bernard Levy -- who was also temporarily running SFR -- was squeezed out in a strategy disagreement with the Vivendi's board. The board is seriously considering breaking apart Vivendi's assets, according to people familiar with the matter. Mr. Levy had long been a advocate of keeping Vivendi's diverse set of assets.

SFR, France's second-largest mobile phone provider after France Telecom SA, is reeling from a price war following the entry in January of a fourth mobile-phone operator, Iliad SA's Free Mobile. Vivendi said earlier that it expects SFR's earnings before interest, tax, depreciation and amortization set to tumble 12% to 15% in 2012.

Write to Sam Schechner at Sam.Schechner@wsj.com