The Legislature sent the bill to Brewer on Monday, setting a Saturday deadline for her to sign the bill, veto the legislation or allow it to become law without her signature.
Elimination of the permit requirement would allow Arizonans to forego background checks and classes that are now required to carry a concealed weapon.
The bill would apply to most U.S. citizens 21 or older.
Currently, carrying a hidden firearm is a misdemeanor punishable by up to six months in jail and a fine of up to $2,500.
Alaska and Vermont are the two states that now do not require permits to carry concealed weapons.