Aldridge will undergo further testing before the Spurs determine how much time he misses.
The Spurs declined to go into details about Aldridge's condition, but a league source told ESPN that Aldridge's "health and safety are of the utmost concern" at this time, and that his return to the court is the least of the club's concerns.
"You just think more seriously," coach Gregg Popovich said Saturday when asked how Aldridge's situation compares to other Spurs injuries. 'When somebody says heart, you start thinking a little more possible long-term kind of stuff. That's a little scary.
"But you don't go there yet. They're doing tests. They did some tests today, and they're going to continue it on Monday. He's going to see some people on Monday, and based on what they do and see, we'll figure then what we're really looking at. I don't know right now."
The 6-foot-11, 260-pound Aldridge was diagnosed with Wolff-Parkinson-White syndrome -- an abnormality that can cause a rapid heartbeat -- at the end of his rookie season in April 2007.
He missed the final nine games of that season with the Trail Blazers after the original diagnosis. He also missed time at the start of the 2011-12 season, when there was a recurrence of his condition.
Popovich said Aldridge complained of "feeling odd" during Thursday's game at Oklahoma City.
Spurs point guards Tony Parker (back stiffness) and Dejounte Murray (groin tightness) also missed Saturday night's showdown with the Western Conference-leading Golden State Warriors. San Antonio was already without star forward Kawhi Leonard, who is in the concussion protocol.
"He's just moving through the protocol," Popovich said of Leonard. "We expect and hope he'll be fine by the end of it. But it just takes a while. So the next possible game is Monday, and I'm hopeful he'll be ready for that."
The Spurs (51-14) ended up starting Patty Mills, Danny Green, Kyle Anderson, David Lee and Dewayne Dedmon in a 107-85 win that moved them within a half-game of the Warriors (52-14) for the No. 1 seed in the Western Conference playoffs. San Antonio has won 10 of its past 11 contests.
Golden State was also short-handed after coach Steve Kerr announced Friday that All-Stars Stephen Curry, Draymond Green and Klay Thompson and top reserve Andre Iguodala would be rested, citing health concerns and the Warriors' recent schedule.