In recent years, media mogul Sumner Redstone has battled with his daughter and business partner, Shari, over their differing visions for the family business. Now, as they race to solve its looming debt problems, those divisions threaten to complicate make-or-break discussions about the future of the Redstones’ holding company.

National Amusements Inc., which holds the family’s controlling stakes in CBS Corp. and Viacom Inc., among other things, has been in urgent discussions with lenders to restructure $1.6 billion of debt and escape a December deadline to pay off half of it. Those negotiations are far from over, but one thing is clear: The Redstones will almost certainly need to sell some holdings to bring their mass of loans and notes under control.

As their advisers explore options, two ideas have gained traction, people familiar with the situation say: selling part of the family’s movie-theater chain, which Ms. Redstone runs and has expanded aggressively, or selling a stake in National Amusements, potentially diluting Mr. Redstone’s control, or at least diluting his returns. Both alternatives could cause a tussle between father and daughter.

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