The Somerset left-armer sustained the injury during warm-ups on the final day of their County Championship clash with Hampshire.
It is a blow that comes just a day before England's squad for the two-Test series with Pakistan, which starts at Lord's on May 24, is announced.
A post on Somerset's Twitter account read: "Jack Leach has sustained a broken left thumb. It's too early to know how long he will be out for at the moment."
Leach had been one of the positive aspects of England's 1-0 series defeat to New Zealand, bowling well as he took two wickets for 113 runs at an economy rate of 2.21 in the drawn second Test that marked his debut.
Headingley hosts the second and final Test with Pakistan on June 1. The second Test series of the English summer starts on August 1 as England face India in five matches.