James had a strained right elbow during the Cavaliers' playoff run.
Bosh had surgery on a facial fracture sustained in early April. He missed the Raptors' final five games of the regular season. He also has missed games with soreness in both knees.
James passed a physical before signing a six-year, $110 million contract and hasn't had any problems with the elbow in workouts with the Heat staff, the newspaper reported.
"There was some soreness there, and there was some treatment there," Riley told the Herald, referring to James' elbow. "It was a real injury, based on what we found out. But it's all fine now. He's getting up a lot of shots and he doesn't look like he's having any after-effects from it at all."
Riley told the newspaper a plan is in place to "monitor and maintain" Bosh's knees.
"He's healed, he's healthy and he's ready to go," Riley said of Bosh. "We'll monitor and see how it goes."