The news comes on the same day ESPN reported that Josh Gordon is facing a season-long suspension for another positive drug test.
Burleson's fracture is on the same arm that he broke last season in Detroit, where he tried to stop pizza from falling off the passenger seat next to him. Burleson's newest fracture, however, is not expected to keep him out of the start of training camp this summer. He is expected to miss the remainder of the offseason program.
Burleson's agent, Ken Sarnoff, posted on Twitter that the veteran needed "a very minor surgery" on the arm.
"He will be 100% for day 1 of camp! Again 100% for camp!" Sarnoff tweeted.
Coach Mike Pettine was asked if Burleson's absence from the last day of the team's voluntary camp was related to an injury.
"He spoke to me," Pettine said. "He's in good standing with us. He's OK."
The Browns said they would not comment on any player's health.
"The league doesn't require us to discuss injuries at this point," Pettine said.
"It's a critical component not to discuss a player's health," Browns GM Ray Farmer added.
Burleson missed almost two months last season in Detroit with the broken arm. In 2012, he missed the last two months with a fractured right leg.
ESPN Browns reporter Pat McManamon and ESPN Lions reporter Michael Rothstein contributed to this report.