Brown, who has been with Washington since 2008, interviewed with the Browns over the weekend.
Brown had been passed over for several general manager positions despite interviews in the last three years. Earlier this offseason, he interviewed with Tampa Bay, but did not receive an offer for that job.
It wasn’t immediately clear what title Brown would hold with Cleveland, but it is believed that it will be a high-ranking position in the front office.
Brown has worked under director of player personnel Scott Campbell, reporting to both Campbell and general manager Bruce Allen.
Brown was instrumental in the Redskins signing free agents under Mike Shanahan, most notably Barry Cofield, Stephen Bowen, and Josh Wilson during the 2010 offseason.
Candidates to replace Brown include front office assistants Doug Williams and A.J. Smith, who have joined the Redskins in the last year and two years, respectively.
Former Redskins offensive coordinator Kyle Shanahan was hired this offseason to direct Cleveland’s offense.
Update, 5:37 p.m.:
“I think it’s a great opportunity for Morocco for his continued development,” Redskins General Manager Bruce Allen said as he left Tuesday’s NFL owners’ meeting in Atlanta. “Him and his wife are great people and we wish them well.”
Allen, as he left the meeting with Redskins owner Daniel Snyder, said the team is about to begin conducting interviews to find a replacement.
“We start interviews this week,” Allen said. “This has been going on. We’ve known this for a few weeks. So we’ve already started setting up interviews.”
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