The 31-year-old is set to miss his side's first four Championship games.
Bresnan played for the champions in their defeat by the MCC last week.
"He will now start an intense period of rehabilitation on his torn calf and we hope to have him back fit and strong during the middle of May," physio Kunwar Bansil told the club website.
Yorkshire and England all-rounder Tim Bresnan will miss the opening six weeks of the County Championship season with a torn lower calf.