If added to the 40-man roster, the 35-year-old would get a one-year contract paying a $1.25 million salary while in the major leagues.
He was catching during Oakland's opening spring training workout.
"He's a well-respected veteran coming off a really good year," Oakland executive vice president of baseball operations Billy Beane said. "We're lucky to make that kind of addition."
Jonathan Lucroy left as a free agent for the Los Angeles Angels, and Chris Herrmann and Josh Phegley are the only catchers on Oakland's 40-man roster. The team also is high on Sean Murphy, a third-round pick in the 2016 amateur draft.
Hundley liked what he saw of the A's, who won 97 games last season and earned a berth in the AL wild-card game, which they lost to the New York Yankees.
"Playing against them, I've always respected the way their players play the game, and always from afar, there are organizations you look at and say, 'I'd like to play for them at some stage of my career,' and this is definitely one of them."
Hundley hit .241 in 96 games with the San Francisco Giants last season.
"He's one of those quarterbacks in the field," Oakland manager Bob Melvin said. "He's a big pickup for us. I know he's excited about being here. You always want a veteran guy, especially in that job, who you have the communication with."