The 40-year-old left the field on Thursday after sustaining the injury during the first day of the match against Somerset at Old Trafford and a "mild groin strain" was diagnosed.
"His fitness will be assessed nearer the time of the test match against Ireland, which takes place at Lord's starting on Thursday June 1, 2023," an ECB statement said.
Anderson and Stuart Broad (36) form the most prolific wicket-taking partnership in test history and the hosts will be desperate to have them bowling in tandem when they play Australia in five matches starting in mid-June.
England are already sweating on the fitness of Jofra Archer for the series after the paceman was ruled out for the remainder of this season's Indian Premier League (IPL) last week due to a nagging elbow injury.
