ESPN The Magazine's Buster Olney first reported the claim.
Bell will become a member of the Giants' bullpen if San Francisco and San Diego can agree to terms on a trade. San Diego has 48 hours to decide whether to trade the three-time All-Star, allow him to be claimed or pull him back from waivers.
Appearing on ESPN Radio's "Doug Gottlieb Show" on Wednesday afternoon, Bell said his agent confirmed that the Giants had claimed him and that now it's a "waiting game."
The Giants continue their series against the Padres on Wednesday night. Bell earned his 35th save in the Padres' victory Tuesday night.
"If I'm still a Padre I'm going to try to beat the Giants. If things change, then things change," Bell said Wednesday.
Bell said he was surprised to be claimed on waivers.
"I knew there was a chance for waivers but I didn't think that would happen," he said.
Giants general manager Brian Sabean might have claimed Bell to block NL West-leading Arizona -- under first-year general manager and former Padres GM Kevin Towers -- from landing Bell. Towers traded for Bell when he was the Padres' general manager.
Baseball sources told ESPN The Magazine's Buster Olney that outfielder Jason Kubel and designated hitter Jim Thome, both Minnesota Twins players, and Chicago Cubs first baseman Carlos Pena also were claimed on waivers Wednesday.
Sources said the New York Yankees claimed Pena. Chicago has 48 hours to decide whether to trade Pena, allow him to be claimed or pull him back from waivers. It is not known at this time which teams claimed the two Twins players.
Brian Wilson, the Giants' closer, is currently on the disabled list with inflammation in his right elbow. There is no timetable for when Wilson will get on a mound, though there's a chance he could return in early September for the stretch run. Wilson has declined to discuss his injury.
The defending World Series champion Giants trail the first-place Diamondbacks by two games in the NL West.
Asked if Bell would still be with the club after the deadline, Padres manager Bud Black said he would be "extremely shocked" if his closer wasn't.
"Albert Pujols was put on waivers. Justin Verlander was put on waivers. I'm sure they got claimed, too, right?" Black said. "I don't know, but every general manager I've known tells me that every player has put been put on waivers."
Bell, too, was already was making plans for the team's series at Arizona beginning Friday.
"I feel like I'm going to Arizona with the Padres and going to do fantasy football on our off day and hang out by the pool and go see a couple movies and go sweat my butt off when I go run in Arizona for Friday's game," he said.
The home team is certainly familiar with the right-hander.
Bell converted his 35th save in 39 opportunities in Tuesday's 7-5 win at San Francisco. He also was chosen by Giants manager Bruce Bochy for the All-Star game, famously sliding onto the infield grass before the NL manager handed him the ball in relief.
The 33-year-old Bell learned that the Giants won the claim on him when a radio station called him for an interview. After a summer filled with trade talk, Bell was hardly surprised.
"I feel pretty honored. They're the world champs, they want me to be part of their organization," he said. "But nothing has happened right now. I'm a Padre, and I'm pretty happy about that."
San Francisco's stellar bullpen is certainly fractured.
There is no timetable for when Wilson will get on a mound and test his inflamed right elbow, although there's a chance he could return in early September for the stretch run. Wilson has declined to discuss his injury. He led the majors with 48 saves last season, while Bell had 47.
Bell said if he is traded do the Giants, it will still be Wilson's bullpen.
"If I end up going to the San Francisco Giants, it's Brian Wilson's bullpen, it's not mine. I just want to be a part of that," he said.
The Giants are more confident that setup man Sergio Romo, also on the disabled list with an elbow injury, will be back soon. Bochy said he hadn't talked to Sabean yet and wouldn't speculate about the team's plans.
"That happens around baseball, so it just happened to make news," Bochy said. "That's what it is right now. When that happens, a club claims somebody, some times those two clubs talk and make a deal. I'm not talking about this as much as what happens with other deals."
Bell has been clear all season that he enjoys playing for San Diego and doesn't want to go anywhere.
While he will likely decline arbitration to become a free agent after the season, the Padres wouldn't lose Bell for nothing. They would likely get compensatory draft picks if he walks.
Bell also has kept open the possibility he would re-sign with San Diego, even if he landed in San Francisco -- or somewhere else -- by the end of the week. In the meantime, he couldn't help but poke fun at how odd it would be to switch clubhouses in the middle of a quick two-game series.
"It would be easy. It wouldn't be that far," he said, smiling. "I know where the locker room is. It would be a little weird just because we've had so many battles over the years."
Information from The Associated Press was used in this report.