The 63-year-old made over 500 appearances for the blue half of Glasgow between 1965 and 1982.
In a statement, the club said: "The club will support Sandy and his family in any way it can throughout his treatment and we ask that the media respect his privacy and indeed the privacy of his family.
"Sandy is a true Rangers legend in every sense of the word and has the admiration and respect of every one of his colleagues at Ibrox and Murray Park."
Jardine, who won the 1972 European Cup Winners' Cup, said: "I would like to thank everyone for their best wishes and kind support."
The defender also played 38 times for Scotland.