The 33-year-old, the world's number one ranked Test bowler, is set to miss Lancashire's next two County Championship games.
England begin their four-match series against Pakistan at Lord's on 14 July.
Anderson suffered a stress fracture in the right shoulder blade during the third Test against Sri Lanka.
Paceman Anderson played a key role in the recent Test series against Sri Lanka, which the hosts won 2-0.
England's leading wicket-taker in Tests took 21 wickets across the three matches, including 10 as the hosts won the opener at Headingley by an innings and 88 runs.
He replaced team-mate Stuart Broad at the top of the International Cricket Council bowling rankings after taking 8-94 in the second match of the series.
The England and Wales Cricket Board (ECB) said Anderson's availability for the Pakistan Test would be "determined following ongoing management and review by both the ECB and Lancashire's medical teams".
Anderson missed the last two Tests of England's 2015 Ashes win with a side strain, before a calf problem forced him to sit out the first Test against South Africa in December.
A short statement from Lancashire read: "The club wishes James well with his recovery and hopes to see him back in action soon."
Durham all-rounder Ben Stokes could replace Anderson after returning to competitive action on Friday after injuring a knee during the Sri Lanka series.
England fast bowler James Anderson is a doubt for the first Test against Pakistan next month after injuring his right shoulder.