Jardine, 63, who was capped 38 times for his country and still works for Rangers in a public relations capacity, has recently been diagnosed with the condition.
He made over 500 appearances for the club as a player between 1965 and 1982, before making more than 200 for Hearts, who he would later manage.
Jardine, who won the 1972 European Cup Winners' Cup with Rangers said: "I would like to thank everyone for their best wishes and kind support. As you can appreciate this is a difficult time for me and my family and I ask that you respect our privacy."
A statement on the Rangers website added: "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 has the admiration and respect of every one of his colleagues at Ibrox and Murray Park.
"There is little doubt he will fight this disease in the same way he fought bravely in every match he featured as a player, and our thoughts are with him and his family at this difficult time."