NFL Network, which first reported the news, reports it's a two-year deal worth $17 million.
With the Chargers, Joseph fills a much-needed role on the interior of the defensive line following the departure of veteran nose tackle Brandon Mebane.
The Chargers released Mebane, along with veteran linebacker Thomas Davis, last week to create $9.5 million in salary-cap space as they pursued Tom Brady, who is now expected to sign with the Tampa Bay Buccaneers.
Joseph became a free agent on March 13 when the Vikings released him to free up money under the salary cap.
Minnesota saved $10.55 million against the cap by cutting Joseph, whose $12.95 million cap hit made him one of the highest-paid defensive tackles in the league.
Joseph, 31, was the first marquee free-agent signing for Mike Zimmer after he was named Minnesota's head coach in 2014. Joseph won a Super Bowl with the New York Giants in February 2012 and signed a five-year deal with the Vikings as a free agent two years later. Joseph then signed a four-year extension worth $50 million ($31.5 million guaranteed) in 2017.
Known for his strength against the run, Joseph had been a staple along the Vikings' defensive line for the past six years. He underwent surgery on his right knee last November to repair a meniscus and returned to action for a Week 13 game at Seattle.
Joseph's playing time decreased in 2019 due to injuries, which made way for an increase in snaps for backup nose tackle Jaleel Johnson and rookie Armon Watts. Minnesota finished the regular season ranked 20th in rushing defense and struggled against the run in the playoffs.
ESPN's Lindsey Thiry and Courtney Cronin contributed to this report.