He said he favors laws that would prevent people with major mental health issues from purchasing firearms.
"If you've been identified with a serious mental health challenge -- maybe you've been in the criminal justice system and your adjudication was withheld and under normal circumstances you can get a gun but you have evidence of a mental health illness, a serious problem -- I don't think you should be able to get a gun," the Republican presidential candidate said at a town hall in Dubuque, Iowa.
"And I think the NRA and other groups would support that notion," he added.
Indeed, the National Rifle Association backed legislation proposed by Sen. John Cornyn of Texas that called for incentivizing states to send records on people with serious mental health issues to the federal background check system -- an issue that gets tricky because of medical privacy laws.
However, such proposals have seen opposition from the staunchest of gun rights supporters who argue that they're too restrictive, as well as from some mental health advocates worried about stigmatizing people who struggle with mental health problems.
In the wake of last week's Planned Parenthood shooting in Colorado, Bush has called for addressing mental health reform rather than cracking down on guns.
"These are completely deranged people, and they have severe mental health issues. In every case there was the possibility of intervening but it didn't happen," he said. "Either there was no knowledge at all, or there was peripheral connection to the mental health network but ... there was no follow through."
But gun control, he added, isn't the right solution.
"The impulse to do something whenever we have these tragedies inevitably impacts the 99.99% of people that use firearms appropriately," he said in Dubuque. "The net result is you don't solve the problem; you're taking people's rights away."
Asked later in the day to name specific gun control proposals that would interfere with Second Amendment rights, Bush struggled to name any restrictions.
"There's all sorts of ones, and I don't agree with them," he told reporters after a town hall in Waterloo, Iowa. "I don't think that Washington is the source of solution for this."
Asked if he was talking about expanded background checks, Bush again didn't mention a specific proposal.
"It's the ones that, that everybody brings up. The whole myriad of them -- starting with President Obama's ideas that were rejected in the Congress," he said, presumably referring to the expanded background check measure that failed to advance in the Senate in 2013.
Bush said he was confident that Republicans could get behind proposals that would limit access to firearms for people who have mental health problems, but he acknowledged that it's a complex issue.
"How do you deal with the questions of privacy? There's a lot of complicated elements of this for sure," he said.