The two teams are currently reviewing medical records of the players involved in the deal, sources told ESPN's Jerry Crasnick.
Familia was held out of Friday night's 7-5 win over the New York Yankees, with manager Mickey Callaway saying he was told midgame not to use Familia because "there could be a potential trade happening."
"It's tough for me," said a somber Familia, who signed with New York as a 17-year-old in 2007. "I love this team. I love my teammates. I have good memories here."
An All-Star in 2016, Familia has 123 saves for the Mets.
"I had a really good time here. It's tough, man," he said.
Familia, 28, is eligible for free agency next offseason. He is 4-4 with 17 saves and a 2.88 ERA in 40 appearances for the Mets this season.
Injuries have slowed Familia in each of the past two seasons. He missed more than a month of this season with a sore right shoulder and more than three months last season with a blood clot in the same shoulder.
In addition, Familia began the 2017 season serving a 15-game suspension for violating MLB's domestic violence policy, stemming from a 2016 incident with his wife.
The Associated Press contributed to this report.
Athletics' pursuit of Familia indicates postseason seriousness
Buster Olney breaks down the implications of Oakland working out a trade with New York for closer Jeurys Familia.