Leach was among in the 16-man squad for the first two Tests announced on Saturday, with England set to name a replacement "in due course" according to an ECB statement.
Leach has been ever-present under Ben Stokes, with the captain trusting him to fulfil a more attacking role in the side. In 13 Tests, he had taken 45 wickets at 38.22, including a maiden ten-wicket haul in the win over New Zealand at Headingley last summer.
The Somerset left-armer is also the only English-qualified spinner to have taken more than 10 wickets in the County Championship this season. Candidates to come into the squad could include Rehan Ahmed, Will Jacks and Liam Livingstone, all of whom made Test debuts in Pakistan over the winter, as well as the previously capped Dom Bess and Liam Dawson.
Leach picked up 3 for 35 in the first innings against Ireland, but struggled for breakthroughs on the third day as Andy McBrine and Mark Adair put on a 163-run stand for the seventh wicket to help avert an innings defeat.
Stokes also turned to Joe Root's offspin at Lord's, and one option for England could be to select another quick bowler and rely on Root for fill-in overs.
England have already lost the services of Jofra Archer, following the recurrences of his elbow problem, and did not consider either James Anderson or Ollie Robinson for the Ireland Test due to minor injuries. Olly Stone is also currently sidelined, while Jamie Overton is recovering from a back stress fracture - although England's pace stocks were buoyed this week by a five-wicket haul on debut for Josh Tongue.