Crowley, who wrestled the Ireland number 10 jersey from Leinster rival Sam Prendergast for the final three games of this year's Six Nations Championship, was assessed by specialists for a lower limb vascular injury, Munster said on their website.
"Jack Crowley is a very important player for Munster and Ireland with a big future in the game and ensuring he gets back to 100% is the priority," the Irish province's head coach Clayton McMillan said.
Ireland face Argentina, Fiji and South Africa at home in the new tournament between northern and southern hemisphere teams that has replaced the traditional mid-year and November tours.
Andy Farrell's side defeated Australia and Japan but were well beaten by New Zealand in their opening games in July.
The 26-year-old outhalf will also at a minimum miss Munster's first two games of the European Champions Cup and the opening phase of the United Rugby Championship.