NEW YORK, May 31 (Reuters) - The U.S. Federal Bureau of Investigation and the Securities and Exchange Commission are investigating possible insider trading involving billionaire investor Carl Icahn, golfer Phil Mickelson and Las Vegas gambler William Walters, a source familiar with the matter said.

Federal investigators are looking into whether Mickelson and Walters may have traded illegally on private information provided by Icahn about his investments in public corporations, the source told Reuters, confirming reports on Friday.

Icahn, a legendary activist investor, told Reuters that he was unaware of any investigation and said that his firm always followed the law. He acknowledged a business relationship with Walters but said that he did not know Mickelson personally.

"I am very proud of my 50-year unblemished record and have never given out insider information," he said.

Walters and Mickelson play golf together, the source familiar with the investigation told Reuters.

Mickelson said in a statement, "I have done absolutely nothing wrong. I have cooperated with the government in this investigation and will continue to do so."

"I wish I could fully discuss this matter, but under the current circumstances it's just not possible," said the statement, issued through his manager.

The Wall Street Journal cited Glenn Cohen, Mickelson's lawyer, as saying the golfing legend was not a target of the federal probe.

Walters did not respond to requests for comment. Spokespeople for the FBI and the SEC declined to comment.

None of the three men have been accused of any wrongdoing, the source told Reuters.

YEARS OF INVESTIGATING

The investigation began three years ago according to the source. It is the latest to emerge from a multi-year crackdown on insider trading by U.S. authorities.

The investigation centers on suspicious trades in Clorox Co by Walters and Mickelson as Icahn was trying for access to the board of the consumer products company in 2011, the New York Times reported, citing people briefed on the probe.

Icahn had accumulated a 9.1 percent stake in Clorox in February 2011. In July, he made an offer for the company that valued it at above $10 billion and sent its stock soaring.

Investigators were also looking into trades that Mickelson and Walters made related to Dean Foods Co, the Journal cited the people as saying. The New York Times cited people briefed on the investigation as saying that in that particular case, investigators are looking into trades placed around 2012 just before the company announced quarterly results.

Those trades appeared to have no connection to Icahn, the newspaper added. Icahn told Reuters he had never purchased shares nor been involved with Dean.

MAKING HEADWAY

Federal prosecutors in Manhattan are handling the inquiry in conjunction with the FBI and the SEC, the New York Times reported. Since August 2009, Manhattan U.S. Attorney Preet Bharara's office has convicted 81 people of insider trading, either at trial or via guilty pleas, with no acquittals.

The current investigation made little headway initially, the Times reported. Investigators are searching phone records, seeking to determine whether Icahn had spoken to Walters before the trades, the Times cited anonymous people briefed on the probe as saying.

And about a year ago, FBI officials approached Mickelson at Teterboro Airport in New Jersey, asking the celebrity golfer to discuss his trading, the Times cited its sources as saying.

Icahn, a prolific tweeter and vocal critic of some of America's largest corporations, habitually broadcasts his thoughts on corporations and, occasionally, stock positions he has taken. The billionaire took to Twitter earlier this year when he pushed Apple Inc's board to expand the iPhone maker's buyback program, and several times tweeted when he had increased his position in the tech company.

During a campaign to get eBay Inc to hive off its fast-growing PayPal division, he openly criticized the Internet retail giant's board. But the activist investor has tweeted just once in the past three weeks.

Even if Icahn did leak information about his plans regarding Clorox, it may not necessarily have violated the law. Insider trading regulations prohibit trading based on material, nonpublic information obtained from someone who breached a fiduciary or confidentiality duty by disclosing it.

As a non-board member, Icahn owed no duty to Clorox shareholders. It is possible, though by no means definite, that Icahn owed a confidentiality duty to his own shareholders.

Mickelson, 43, is currently playing at the Memorial Tournament in Dublin, Ohio ahead of the June 12-15 U.S. Open.

The American player clinched his fifth major title at last year's British Open. He would complete a career grand slam of the four majors by winning the U.S. Open, where he has been runner-up a record six times. (Additional reporting by Mark Lamport-Stokes; Writing by Edwin Chan and Frances Kerry; Editing by Peter Henderson and Louise Heavens)