Suns coach Jeff Hornacek said Bledsoe is expected to miss about six weeks.
Bledsoe suffered the non-contact injury Saturday night in a loss to the Philadelphia 76ers.
He also suffered a torn meniscus in his right knee during the 2013-14 season, an injury that required him to undergo midseason surgery and miss 33 games.
Bledsoe's latest injury comes at a challenging time for the Suns, who have lost 15 of their past 20 games to slide back in the Western Conference standings and put coach Jeff Hornacek's job in jeopardy.
Bledsoe was having a career year, averaging 20.4 points, 6.1 assists and 2 steals per game.