Staff at Tesco Express stores in Bath Road, Southcote, and Church Street, Caversham, plus the Tesco superstore in Napier Road, Reading, were each fined £80.
Now John Ennis, ward councillor for Southcote, is pressing licensing councillors at Reading Borough Council to review licences to sell alcohol at the stores.
If they agree to do so, it will be the first action of its kind since changes to licensing laws came into effect in November last year under the Licensing Act 2003. Lacors, the local government regulator, said that a review of store licences was an option that local councils could resort to in such circumstances.
Tesco said: "We view these failures very seriously. As a responsible retailer, we do have very strict procedures for dealing with the sale of alcohol to underage customers."
It said it drilled staff at Tesco and One Stop stores not to sell alcohol to anyone who looked younger than 21.
The Home Office is to run a fresh wave of test purchasing stings in England and Wales from June 8, with Scotland's first test purchasing initiative due to kick off mid-June.