Clary injured a ligament in his left ankle during Sunday's 32-3 victory at Denver.
Runyan hasn't played since the end of last season. He wasn't re-signed by the Philadelphia Eagles after having microfracture surgery on his right knee.
Runyan never missed a game in nine years with the Eagles, starting 192 consecutive regular-season games. He turns 36 on Friday.
Chargers general manager A.J. Smith and coach Norv Turner didn't immediately return calls seeking comment Tuesday night. However, Smith did issue a statement in the news release announcing the move, saying: "We are very fortunate that a player of Jon's experience and ability was available at this time. He is an outstanding competitor, and I have always admired the physical, nasty approach he brings to the field every game. He will be added to the mix and we will move forward."
Earlier this month, Runyan said he's considering running for Congress in New Jersey's 3rd District.
The Associated Press contributed to this report.