A long-running mystery over who killed the wife and son of Alex Murdaugh, an heir to a powerful legal dynasty in South Carolina, could be headed for a resolution this week as the police indicated they planned to indict Mr. Murdaugh in the murders, one of his lawyers said on Tuesday.

Mr. Murdaugh had already been facing an array of charges of fraud and theft that resulted in his arrest in September 2021 and a wave of subsequent indictments. But no one had been formally accused in the deaths of Maggie Murdaugh, 52, and Paul Murdaugh, 22, who were fatally shot at the family’s rural hunting estate in Islandton, about 65 miles west of Charleston.

One of Mr. Murdaugh’s lawyers, Jim Griffin, said state police investigators had informed some of Mr. Murdaugh’s relatives that they planned to seek an indictment against Mr. Murdaugh for the murders this week. A spokeswoman for the South Carolina Law Enforcement Division, which has been investigating the case, would not confirm the lawyer’s account.

It was Mr. Murdaugh, sounding deeply shaken, who called 911 on the night of June 7, 2021, and told a dispatcher that he had returned home to find the bodies of his wife and son lying near a dog kennel on the family’s secluded estate, which sits far back from the road at the end of a winding driveway.

Mr. Murdaugh, 54, is a fourth-generation lawyer whose father, grandfather and great-grandfather each served as the top prosecutor in a district that reaches across a wide swath of the South Carolina Lowcountry region.

But in the months that followed the double-murder, Mr. Murdaugh’s life unraveled drastically.

The law firm founded by his ancestors reported that Mr. Murdaugh had been stealing money from the firm and fired him, after which, the police said, he concocted a bizarre scheme to have one of his distant cousins shoot him on a deserted road. The cousin, Curtis Edward Smith, fired a shot that did not hit Mr. Murdaugh, and Mr. Murdaugh was later charged with fraud while Mr. Smith was charged with assisted suicide and other crimes.

Last month, prosecutors filed additional charges against Mr. Murdaugh and Mr. Smith, accusing them of laundering millions of dollars over several years as part of a distribution network for the painkiller oxycodone.

Mr. Murdaugh has also been charged with stealing millions of dollars from clients and other people, including from the family of a housekeeper for the Murdaugh family who died after falling on the steps of their home early one morning in 2018. Mr. Murdaugh admitted last month that he had wrongly pocketed the settlement money and owed $4.3 million to the sons of the housekeeper, Gloria Satterfield. After having been briefly released following his initial arrest, Mr. Murdaugh returned to jail in October on charges related to the settlement with the housekeeper’s family.

On Tuesday, the Supreme Court of South Carolina issued an order disbarring Mr. Murdaugh over his theft from the Satterfield family.

Mr. Griffin said Mr. Murdaugh continued to proclaim his innocence in the deaths of his wife and son.

At the time he was killed, Paul Murdaugh had been facing charges of boating under the influence after he was accused of driving a boat into a bridge, killing Mallory Beach, a 19-year-old passenger, after a night of drinking. Ms. Beach’s family had sued the Murdaughs, and their lawyer had been trying to force Alex Murdaugh to provide details about his finances before his wife and son were killed.

The killings turned intense scrutiny back to the boat crash — including over whether Paul Murdaugh had received favorable treatment from the police — and raised new questions about the Murdaugh family.

The state police opened an investigation into the death of Ms. Satterfield and, last month, said they planned to exhume her body. Investigators have also been re-examining the death of a 19-year-old man, Stephen Smith, who was found dead along a road about 10 miles from the Murdaugh home in 2015. No one has been charged in those deaths, which were both considered accidents.