The idea began with Bob Wright, a former NBC chairman and longtime Trump confidant, who first proposed creating a government research arm, called the Health Advanced Research Projects Agency (HARPA), to tackle health problems.
Mr Wright’s team put together a three-page document titled “Stopping Aberrant Fatal Events by Helping Overcome Mental Extremes”, which links mass shootings to mental health.
Such a view is highly controversial, with experts believing the link between the two is weak.
The document urges the government to explore whether technology could help combat violence.
It suggests that new technology could detect when people with mental health difficulties are about to become violent.
Marisa Randazzo, former chief research psychologist for the US Secret Service, told The Washington Post that the proposal was concerning.
The technology required is yet to be developed and tracking people would also infringe their civil liberties.
Ms Randazzo told the newspaper that the initial premise was itself flawed.
“Everything we know from research tells us it’s a weak link at best,” she said.
The El Paso massacre was carried out by a man with racist motivations but conservatives in the US have repeatedly suggested that mental health was to blame.
“I don’t want people to forget that this is a mental health problem. I don’t want them to forget that, because it is. It’s a mental health problem,” Donald Trump said, following the El Paso massacre and another mass shooting in Dayton, Ohio.
“It’s the people that pull the trigger, it’s not the gun that pulls the trigger,” he continued, echoing a favourite Republican talking point.
Statistics show that other countries with stricter gun control measures suffer far lower levels of gun violence than the US.
The president has avoided questions about gun control since the shootings.
In August Mr Trump was asked if he would support banning high-capacity magazines, such as the one used by a shooter in Dayton, Ohio.
He dodged the question, and instead ranted about the closure of hospitals for people with mentally health problems.