The amendment would allow lawmakers to convene after the end of the session on bills the governor rejected.
That would allow them to enact a bill despite a veto if they have a two-thirds majority vote from each house.
Sen. Steve Vick, a Republican from Dalton Gardens, said Idaho is one of only seven states that don't have that ability.
He calls it a way to maintain the state's checks and balances.
But other lawmakers worry it goes too far.
Sen. Elliot Werk, a Boise Democrat, said the amendment could be a blow to the balance of power.
Gov. C.L. "Butch" Otter is the only person who can call the Legislature to convene.