The sad news was confirmed on the Scottish club's website late on Thursday evening, along with a message of condolence from manager Ally McCoist.
"Rangers Football Club tonight learned of the tragic news that Claudio Reyna's son Jack has sadly died after a battle with cancer," the statement read.
"The thoughts of everyone at the club are with Claudio and his family at this extremely difficult time."
Ex-United States international Reyna spent two years in Glasgow from 1999, when Jack was born, to 2001.
McCoist said: "We have just heard the sad news that Claudio Reyna's son Jack has lost his battle against cancer.
"On behalf of everyone at Rangers I would like to offer my sincere condolences to Claudio and his family at this difficult time.
"Losing a child at just 13 is a tragedy and puts football into perspective."