Jack underwent an operation to fix a persistent calf problem on Wednesday morning.
The 29-year-old has not played since February 21 and will miss Scotland's European Championship campaign.
But the former Aberdeen player is confident of being back in time for the start of next term and is already looking ahead to defending Rangers' Scottish Premiership title.
He wrote on Instagram: "Just letting everyone know I had a successful operation yesterday morning and I'm now on the road to recovery.
"Can't wait to be back for the start of next season and getting back doing what I love. Thanks to everyone for the messages it's much appreciated. All focus now is on 56!"
Rangers will kick off their Premiership season on the weekend of July 31-August 1 before their Champions League qualifying campaign begins several days later.