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The hedge fund owned 5.1 percent of Staples last month when it boosted its stake in Office Depot to nearly 10 percent.
A merger will result in synergies of at least $2 billion, more than doubling operating profits, Starboard's founder and Chief Executive Jeffrey Smith wrote in a letter to Staples' Chief Executive Ronald Sargent on Tuesday.
"If it becomes clear to us that you have no intention of seriously pursuing this unique and highly attractive opportunity, it would be a clear sign that significant leadership change is needed at Staples," Smith said.
Staples, the No.1 office supplies chain, said it has met Starboard Value to discuss their ideas, and carefully considers all actions that would create shareholder value.
A merger would help fend off intense competition from online retailers such as Amazon.com Inc <AMZN.O> and big-box chains such as Wal-Mart Stores Inc <WMT.N> that sell the same core office supplies, such as paper and ink toner, for less.
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However, Staples would reject Starboard's proposition due to antitrust concerns, the Financial Times reported, citing people familiar with the matter. (http://on.ft.com/1ynnIIB)
Office Depot and Staples were not immediately available to comment on the FT report.
A combination of the two biggest remaining U.S. office supply retailers will face a long review by the U.S. Federal Trade Commission (FTC), Michael Keeley, partner at law firm Axinn, Veltrop & Harkrider LLP, told Reuters.
The FTC would either outright clear or block such a deal, and would not ask for store divestitures, Keeley said.
Regulators nixed Staples' attempt to buy Office Depot in 1997, citing antitrust concerns.
The FTC approved Office Depot's $976 million acquisition of OfficeMax in 2013 without the need to close stores, citing increased competition in the office supply industry.
The magnitude of value created from such a combination would far exceed anything that either company could achieve on a standalone basis, Smith said.
Staples shares were down 5 percent at $16.45 in late afternoon trading on Tuesday, while Office Depot shares were down 4.1 percent at $7.71.
(By Sruthi Ramakrishnan and Yashaswini Swamynathan; Additional reporting by Devika Krishna Kumar, Editing by Joyjeet Das and Savio D'Souza)