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Phil Mickelson, the famed golfer, did not trade in the shares of Clorox just as the billionaire investor Carl C. Icahn was mounting an unsolicited takeover bid for the company in 2011, say four people briefed on the matter.

Recent reports in The New York Times and other news organizations said that Clorox was among the stocks that federal authorities were examining as part of a two-year investigation into well-timed trades made by Mr. Mickelson and the sports gambler William T. Walters. Initially, some investigators pursued a theory that Mr. Icahn shared private details of his Clorox bid with Mr. Walters, who then traded on the information and passed on the tip to Mr. Mickelson.

Although Mr. Icahn and Mr. Walters remain under investigation over Clorox, the F.B.I. and the Securities and Exchange Commission have found no evidence that Mr. Mickelson traded Clorox shares. The overstated scope of the investigation came from information provided to The Times by other people briefed on the matter who have since acknowledged making a mistake.

The F.B.I. is pursuing a criminal investigation, while the S.E.C. is running a parallel civil inquiry.

Mr. Mickelson and Mr. Walters, as previously reported, still face an investigation over separate well-timed trades they made in Dean Foods in 2012 just before the company’s stock soared. Those trades generated more than $15 million in proceeds for Mr. Walters and nearly $1 million for Mr. Mickelson, one of the four people briefed on the matter said. Mr. Mickelson has denied any wrongdoing.

Mr. Mickelson, a three-time winner of the Masters golf tournament who is set to compete on Thursday at the United States Open, received some assurances from the government itself. An F.B.I. agent, two of the people briefed on the matter said, has informed Mr. Mickelson that the government is seeking information from him about Mr. Walters and has no plans to criminally charge him. That did not stop the government, which has not accused anyone of wrongdoing in the investigation and ultimately might not file either a criminal or a civil case, from making a pitch to secure Mr. Mickelson’s cooperation. F.B.I. agents have twice approached him out of the blue, the people said, once at an airport and once on a golf course at a tournament.

The new details, provided in the interviews with the people briefed on the matter, indicate that Mr. Mickelson’s ties to the investigation are weaker than previously reported. The details may also raise questions about the government’s decision to deploy what appear to be unusually aggressive tactics in the investigation, particularly when the F.B.I. agents publicly approached Mr. Mickelson even though he is known to have a lawyer and a sports agent.

At the airport in September, one of the people said, Mr. Mickelson had little to offer. He pledged to cooperate but explained that he did not know Mr. Icahn and had no clue that any stock tip he received from Mr. Walters might have been improper.

When confronted at a golf tournament in Ohio last month, Mr. Mickelson instructed F.B.I. agents to “speak to my lawyers,” according to a nearby observer.

Mr. Mickelson has hired Gregory B. Craig, a partner at the law firm Skadden, Arps, Slate, Meagher & Flom who was previously White House counsel under President Obama.

Several former federal prosecutors who worked on insider trading investigations, speaking on the condition of anonymity to avoid antagonizing law enforcement officials, questioned the decision to approach Mr. Mickelson after he finished a round of golf at the Memorial Tournament in Dublin, Ohio. The former prosecutors speculated that the government wanted to embarrass him into cooperating against Mr. Walters.

“If this were the only effort they made at that time to contact him, having it be so public could be seen as an aggressive move,” said Daniel C. Richman, a criminal law professor at the Columbia University School of Law.

Mr. Mickelson, Mr. Icahn and Mr. Walters have issued statements denying any wrongdoing. Mr. Mickelson, who said he has “done absolutely nothing wrong,” added “I’ll cooperate as much as I can with the F.B.I.” In an interview. Mr. Icahn said, “I don’t give out inside information,” adding that “for 50 years I have had an unblemished record.”

Richard Wright, a lawyer for Mr. Walters, said his client had not had any contact with the authorities about his trading activity. Mr. Wright said no one from the F.B.I., the S.E.C. or the Justice Department had reached out to Mr. Walters, who is known to be an active stock trader, either before or after the investigation became public.

Mr. Icahn similarly has not received any subpoenas from the authorities, a person briefed on the matter said.

The lack of subpoenas struck some former prosecutors as odd. Once the investigation became public, the former prosecutors said, they would have anticipated authorities sending out subpoenas seeking access to emails and other records. Such documents would most likely be needed to help establish that Mr. Walters and Mr. Mickelson had traded stocks while in possession of inside information.

And even if Mr. Icahn did share secret information about his firm’s Clorox intentions, he may have done so legally. It would be illegal if he breached a duty of confidentiality to his own investors.

Any suspicion that the three — Mr. Mickelson, Mr. Icahn and Mr. Walters — formed a triangle of insider trading has failed to materialize. Nothing in the public record suggests Mr. Icahn and Mr. Mickelson even know each other. Mr. Icahn and Mr. Walters both have acknowledged knowing each other — people who know them say the two men have wagered on sporting events together. But both Mr. Icahn and Mr. Walters insist they have never shared inside information.

Mr. Mickelson and Mr. Walters have participated in golf tournaments together. And outside the world of sports gambling, Mr. Walters, better known as Billy, is a prominent Las Vegas figure as the owner of popular golf courses.

It is unclear why federal authorities initially suspected that Mr. Mickelson may have played a role in the Clorox trades. The S.E.C. began the inquiry into Clorox trading after finding suspicious trades in the stock just days before Mr. Icahn’s unsolicited bid.

Although Mr. Mickelson is not connected to the Clorox trades, he is not in the clear on Dean Foods.

The F.B.I., federal prosecutors in Manhattan and the S.E.C. continue to investigate well-timed trades made by Mr. Mickelson and Mr. Walters in shares of Dean Foods in the summer of 2012, the people briefed on the matter said.

The authorities are pursuing a theory that a source inside Dean Foods gave Mr. Walters a heads-up about the company’s plan to spin off its WhiteWave Foods subsidiary in an initial public offering, though it is unclear what exactly prompted Mr. Walters and Mr. Mickelson to trade.

Shares of Dean Foods surged 41 percent on Aug. 8, 2012, the day after the company officially announced the spinoff in a news release.

The trading in Dean Foods has no apparent connection to Mr. Icahn.