Edgar Bronfman Jr., the former chief executive of Warner Music Group and the heir to the Seagram fortune, submitted a bid on Monday to gain control of Paramount, the media conglomerate that owns CBS, MTV and the movie studio behind such hits as “The Godfather” and “Mission: Impossible,” according to two people familiar with the offer who were not authorized to speak publicly.

The bid is the latest twist in a monthslong saga over control of Paramount, a once-mighty TV and film colossus. In July Skydance, the Hollywood studio founded by the technology scion David Ellison, agreed to buy out Shari Redstone, Paramount’s controlling shareholder, and other shareholders and recapitalize the company. But the deal, in which Skydance and its backers would put up roughly $8 billion, had a 45-day “go-shop” provision that allowed Paramount to seek a better offer.

Mr. Bronfman is taking advantage of that window. He is part of a bidding group that includes the producer Steven Paul, creator of the “Baby Geniuses” movie franchise, according to two people familiar with the matter.

The fate of Mr. Bronfman’s bid should be clearer on Wednesday, when that “go-shop” window expires. A decision by Paramount to extend the time it has to negotiate a deal would signal that the company is taking his offer seriously.

According to one of the sources, Mr. Bronfman has offered $4.3 billion for control of Paramount, a company that was worth roughly $25 billion when it was created by the merger of Viacom and CBS five years ago. The proposal would give Mr. Bronfman ownership of National Amusements, the parent company of Paramount, allowing him to control the media conglomerate.

News of the bid was reported earlier by The Wall Street Journal.

A longtime media executive, Mr. Bronfman is said to be interested in running Paramount as its chief executive if his bid prevails. In recent weeks, he has met with Ms. Redstone to discuss his bid, according to two people with knowledge of the meetings.