The bill by Republican Sen. Gerald Allen of Tuscaloosa would do away with the current requirement to get a concealed carry permit.
The Senate Judiciary Committee on Wednesday adjourned before a public hearing on the bill after running short on time. The committee chairman said the public hearing will be rescheduled for next week
A number of state sheriffs and the members of the group Moms Demand Action came to the committee to oppose the bill. Sheriffs have said the permits are needed for public safety.
Allen has proposed the bill for several years, saying people should be able to exercise their right to bear arms without a fee.