A long-running legal battle between American International Group Inc. and an investment firm led by its former chief executive, Maurice R. "Hank" Greenberg, is due in federal court Monday. Control of more than $4.3 billion is at stake.

At issue are tens of millions of shares Starr International Co. holds in the giant insurer, as well as tens of millions of shares Starr sold in recent years.

Were the insurance giant to prevail, AIG said Thursday that it intends to use any cash it wins to help pay back the debt it owes the federal government, which rescued it from the brink of bankruptcy in September. Any winnings that it gets in stock, currently worth a far smaller amount than the cash Starr has generated from selling other stock, will be used toward employee long-term compensation.

Mr. Greenberg is expected to testify in the trial, before a jury in federal district court in lower Manhattan. Both sides have retained name-brand counsel, David Boies for Starr, and Theodore Wells for AIG. The case could still settle.

For decades when Mr. Greenberg was at AIG's helm, the shares were used to fund a long-term compensation plan for AIG employees. But after he left in 2005, amid an investigation of AIG's accounting, the program ended and Starr, which is privately held, has been using the funds for other purposes, including investments. AIG says it should control the shares.