Sen. John Cornyn (R-Tex.) has a bill he says will help background checks keep guns out of the hands of the mentally ill. (Andrew Harnik/AP)

For a long time, the Republican response to more gun control was to double down on calls for better mental-health services to catch disturbed shooters who shouldn’t have access to guns.

But the renewed focus on gun control in the wake of President Obama’s proposals may just exacerbate existing tensions on mental-health efforts that advocates hoped would gain momentum in 2016.

Republicans are not welcoming Obama’s latest challenge as part of a package of gun-control proposals to “put your money where your mouth is” and support increasing mental-health spending by $500 million. Those arguing for a broader approach to mental-health legislation — one that emphasizes systemic changes to get help to the families and individuals that the system has missed or failed — worry that the hyper-partisan gun-control debate could hurt their cause.

“This bill is not a gun issue, it’s a mental-health issue,” said Rep. Tim Murphy (R-Pa.) of his bill, which seeks to unite federal mental-health programs under a single office, expand treatment services and better enable commitment procedures to target the most severe cases. “I think it diminishes its importance to say this is the counterpoint to gun control. It’s not.”

The gun-control debate is both a blessing and a curse for mental-health reform backers. On one hand, it has heightened the profile of their effort — supporters have occasionally nodded to mass killings where the shooters had mental disorders. But on the other, it has drawn them into a political powder keg.

There are now at least two competing versions of mental-health reform kicking around each chamber, some more closely linked to gun control than others. But all of the proposals have been pulled into the stormy debate over gun control by association.

“We have seen consistently that an underlying cause of these attacks has been mental illness, and we should look at ways to address this problem,” House Speaker Paul D. Ryan (R-Wisc.) said Monday.

Murphy’s bill has dozens of Democratic co-sponsors, despite disputes over how the bill handles the delicate issues of patient privacy and involuntary commitments — issues that have also drawn criticism from the mental-health community along with support for opening the conversation.

That community has been slightly more welcoming of a bipartisan effort from Sens. Bill Cassidy (R-La.) and Chris Murphy (D-Conn.) that tweaks Tim Murphy’s proposal to resolve partisan disputes that had cropped up in the House. It also has bipartisan support — seven of the 14 co-sponsors are Democrats.

But only one bill has had a vote — in a House Energy and Commerce subcommittee, where Tim Murphy’s bill was approved in November on an 18-to-12 vote. While more hearings are expected, neither bill has been scheduled for a full committee markup.

And a competing set of mental-health bills focusing more closely on the intersection of mental health and gun control, drafted by Sen. John Cornyn (R-Tex.) and Rep. Martha McSally (R-Ariz.), are picking up momentum.

Democratic leaders remain staunchly opposed to Cornyn’s proposal, which the Senate majority whip says would improve the ability of background checks to keep guns out of the hands of the mentally ill. Democrats argue that in the process, it would weaken the background check system. Cornyn’s bill has no Democratic support, though McSally’s bill, which is effectively the same legislation, has a handful of Democrats on board.

Cornyn is pushing his legislation as an alternative to Obama’s gun-control measures that Republicans may be able to rally around as mass shootings continue to capture the national consciousness. According to a Cornyn aide, the bill should get a Judiciary Committee hearing next month, and Cornyn is working with that panel and the Health, Education, Labor and Pensions Committee to merge his bill with the Cassidy-Murphy legislation.

Cassidy and Chris Murphy, however, are wary of linking their measure too closely to Cornyn’s bill, which is a lightning rod for cross-aisle criticism in the gun debate. Murphy is against merging the two measures, and a Cassidy spokesman also balked at the idea of close coordination.

“This bill is separate from the Cornyn bill and is moving forward on its own trajectory very well,” said Cassidy spokesman John Cummins.

Cummins added that he thought the pace with which the Cassidy-Murphy bill was moving through Congress was appropriate. But not everyone feels as comfortable with how things are proceeding.

“Quite frankly, I wish executive action could be taken to speed this process up,” Tim Murphy said.

He said there is too little coordination and transparency in the federal government’s approach to addressing mental illness, adding that Obama’s proposed $500 million investment straightaway won’t solve that problem. “That could be good,” he said. “But it depends on where it goes.”