Indiana Gov. Mike Pence raised concerns this past week about a bill requiring an armed "school protection officer" onsite during school hours.
"Decisions that are nearest and dearest to our hearts ought to be made by parents and local school officials," Pence told reporters.
Some states such Texas and Utah already allow teachers and administrators to bring guns to school, though the practice is not common. Just three Texas school boards have granted permission for concealed guns, said state Sen. Dan Patrick, a Houston Republican who is sponsoring legislation to train armed teachers for classroom gunfights.
In Minnesota, where the gun debate is on hold at the Capitol, the small town of Jordan recently decided to place satellite police offices in its public schools. The intent was that the mere presence of police would deter any would-be attackers.
Some ardent guns-rights supporters remain hopeful that stalled legislation still can pass this year.
Texas Rep. Dan Flynn, a Republican co-author of a bill allowing guns on college campuses, said opposition from public universities and big cities has so far kept the measure from coming to a vote. But the Legislature doesn't adjourn until Memorial Day.
"This is still Texas," Flynn said. "And in Texas, the Second Amendment is right up there with mother, God and apple pie."
Associated Press writers Tom Davies in Indianapolis; David Eggert in Lansing, Mich.; Sean Murphy in Oklahoma City; and James Vertuno in Austin contributed to this report.