CBS Corp. and controlling shareholder National Amusements Inc., embroiled in one of the media industry’s most bitter public disputes, may be nearing an end to their feud before it goes to trial next month.

The two companies are weighing a resolution that would let NAI President Shari Redstone appoint new directors to CBS’s board, said the people, who asked not to be identified because the deliberations are private. Redstone would also pledge not to pursue a merger between CBS and Viacom, which is also controlled by NAI, the people said.

While the talks signal some progress and the details of a potential deal are still being negotiated, there’s no guarantee an agreement will be reached, the people said. Representatives for CBS and NAI declined to comment.

CBS and NAI are set to face off in a Delaware court over whether CBS directors, led by Chief Executive Officer Leslie Moonves, can strip NAI of its controlling interest in the company. CBS has opposed a merger with Viacom, concerned that it would mean tying itself to a weaker performer. The two sides also disagreed over who will run the combined entity.

Harassment Allegations

Moonves, 68, faces separate allegations that he sexually harassed several women and oversaw a culture where such incidents were covered up. CBS has said it takes any claims seriously and that independent directors of the company are investigating.

NAI had asked independent directors of CBS and Viacom to explore a merger between the companies. The directors eventually reached agreement on economic terms of a deal, but they couldn’t settle the issue of leadership: CBS had pushed for Moonves and his deputy Joe Ianniello to lead the merged entity, while NAI wanted Viacom boss Bob Bakish to also be part of the team.

But in May, NAI told Viacom directors it was no longer interested in merging the two companies, according to court filings. NAI didn’t “intend to force a CBS/Viacom merger, whether by removing and replacing CBS directors or otherwise,’’ NAI executives said in the Delaware Chancery Court suit. That decision was made before CBS’s board approved a plan to dilute the Redstone’s voting control as a defensive measure against the Viacom merger, NAI said.