The England Twenty20 and one-day international opener is expected to miss six T20 Blast and three County Championship matches during his time on the sidelines.
Lumb sustained the injury while making 68 as Notts scored 387 runs on the final day to beat Middlesex at Trent Bridge last week.
"Michael needs a period of rest, which is a blow at this stage of the season given his record in all formats," Notts director of cricket Mick Newell told the club website.
"It's very important that we allow him time to recover and get him back into the side, knowing that Phil Jaques will only be around for three more weeks."
Jacques will replace Lumb for this week's Twenty20 double header against Leicestershire Foxes and Derbyshire Falcons.