Adams had gone into cardiac arrest about an hour earlier at his family's home, said Marcia Kelley-Clark, chief deputy coroner for Greenwood County.
heart attack, Kelley-Clark said. She said relatives were unaware of any medical condition.
Toxicology tests are being run by the state's Law Enforcement Division, but drugs are not believed to have been a factor in Adams' death. Results probably will not be available for at least two months, Kelley-Clark said.
"He was a terrific young man," NFL Commissioner Roger Goodell said Sunday. "I met him at the draft when he came in, and he stood out.
"He seemed like a very genuine, nice man. He seemed focused on being a good person, not just an NFL player. I was always taken with him."
Adams, a 6-foot-5, 258-pound defensive end, was born June 8, 1983, in Greenwood. He was selected fourth overall in the 2007 NFL draft by Tampa Bay and was traded to the Bears in October 2009 in exchange for a second-round pick in the 2010 draft.
"Gaines was a quiet, humble kid and is far too young to be gone," Tampa Bay cornerback Ronde Barber said. "He had so much potential that had yet to be achieved."