Former Test captain Cook was cleared to play in the opening round of championship fixtures, along with several of his fellow centrally-contracted England stars, and was set to face his international team-mate James Anderson on his first-class return in Division One on Friday.
Cook has already undergone scans, and the injury is not thought to be serious.
But it does mean he will not be able to begin preparing for his first Test under new captain Joe Root against South Africa at Lord's in July until at least next week's Division One match against Somerset at Taunton.
The 32-year-old 140-Test veteran had been cleared to play in the opening round of championship fixtures, along with several of his fellow centrally-contracted England stars.
But a statement published on the England and Wales Cricket Board website read: "Alastair Cook will miss Essex's opening championship fixture against Lancashire starting this Friday due to left hip soreness.
"Cook has been receiving treatment for the problem for several weeks. A scan this week has ruled out serious injury, but he will undertake an injection in the upcoming days to help settle his symptoms.
"Cook will continue his rehab with the Essex medical team."