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Shares of Symantec surged 22 percent in after-hours trading, while Broadcom fell 4 percent.
A deal would expand Broadcom's push into software a year after its $18.9 billion deal to buy U.S. business software company CA Inc. It also follows Broadcom's failed bid to buy Qualcomm Inc.
A Symantec spokesperson said the company does not comment on speculation. Broadcom was not immediately available for comment.
Broadcom's chief executive officer, Hock Tan, has built the company through a series of ambitious deals. As CEO of Avago, Tan pulled off the acquisition of Broadcom for $37 billion, bringing the two companies under the umbrella name of Broadcom.
A Broadcom-Symantec deal would be the second major acquisition of a cybersecurity provider by a chipmaker, following Intel Corp's purchase of California-based McAfee in 2011. However, Intel sold a majority stake in McAfee to investment firm TPG after a failed effort to stake out a major position in the computer security business.
Symantec was an early leader in computer security but is facing growing competition.
The company, which is being investigated by U.S. regulators over an accounting irregularity, has seen a slew of key executive departures since late last year, including CEO Greg Clark, who abruptly stepped down in May. In August last year, Symantec said it would cut 8 percent of its workforce worldwide after it slashed its revenue forecast.
The anti-virus software maker's recent troubles have hit its stock, which as of Tuesday's close has lost over a third of its value from a high of $34.20 in August 2017.
News of the talks between Broadcom and Symantec was first reported by Bloomberg.
(Reporting by Greg Roumeliotis in New York; Additional reporting by Bharath Manjesh in Bengaluru; Editing by Bill Rigby, Cynthia Osterman and Leslie Adler)