Spangenberg, the club's Opening Day second baseman, has been out since April 19, when he sustained the injury, while legging out an infield single.
"In Spangenberg's case, that's been a guy that we looked at, based on his rookie year, to really be a catalyst in our offense this year," said Padres manager Andy Green. "He's been a tough loss for us."
Both Spangenberg and Weeks are running and going through baseball-related activities. They'll likely participate in Arizona League games during the All-Star break.
Given the length of time they've spent on the disabled list, Green said Spangenberg and Weeks will most likely require extensive rehab stints that could last the maximum 20 days.
Weeks, who was initially placed on the disabled list on May 8, was transferred to the 60-day DL Wednesday.
"It doesn't necessarily stall him out or keep him from getting back any sooner," Green said. "It's just the fact that if you go back to the first day he went on the DL, it's going to be 60 days before he's ready to play again. Nothing's changed in his progression. He's in Peoria, hopefully moving forward well."
Moving Weeks to the 60-day DL gave the Padres room on their 40-man roster to claim right-handed reliever Jake Smith off waivers from San Francisco. Smith was immediately optioned to Double-A San Antonio.
Smith posted a 7.08 ERA in 20 1/3 relief innings this year for Double-A Richmond, striking out 26, while walking 23. He had a much better season in 2015, posting a 2.35 ERA with 118 strikeouts and just 21 walks in 84 1/3 innings.
AJ Cassavell is a reporter for MLB.com. Follow him on Twitter @ajcassavell. This story was not subject to the approval of Major League Baseball or its clubs.