In a conference call with reporters, Roberts, 37, said doctors have indicated his prognosis is "very positive" because the cancer was detected early, the Boston Herald reported.
"I expect to beat this and be fully recovered," Roberts said.
The Herald said Roberts, who remains a hero to Red Sox fans for his key stolen base in Boston's 2004 ALCS victory over the New York Yankees, indicated he has begun chemotherapy treatments after feeling a lump in his neck before spring training.
"I'm doing good," Roberts told reporters. "Once the initial shock wore off, I'm at the stage now of 'What can we do to beat this?' My wife's been incredible.
She's been a pillar of strength for our family and I'm feeding off her energy."
Roberts played for the Indians, Dodgers, Red Sox, Padres and Giants in a nine-year career during which he totaled 721 hits, 23 homers, 213 RBIs and 243 stolen bases over 832 games.
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