"Right now, we're getting close to the magic 60 number in terms of our co-sponsors. ... Hopefully, we'll have some more news about that later (this week)," Cornyn said.
Dozens of senators added their names to the bill after a shooter killed 17 people at a Parkland, Fla., high school last month.
After that mass shooting, the White House invited lawmakers from both parties to a meeting to discuss gun safety, during which Trump pitched much broader proposals, including ones that many Republicans dislike.
Cornyn said Trump called him after that meeting, reaffirming his support for Fix NICS. Trump also met with the NRA, which supports Cornyn's bill, that same evening.
Cornyn's plan is one of a handful introduced by lawmakers this week.
Gun safety advocates want Congress to do much more to curb gun violence than Cornyn's bill. But they support it as a first step.
"The Fix NICS Act is a no-brainer -- it should have 100 co-sponsors," said John Feinblatt, president of Everytown for Gun Safety. "However, this modest piece of legislation does not meet the moment, and if it's all that Congress does, it would be an inexcusable failure of leadership."
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Sen. Ted Cruz, R-Texas, who is not a sponsor on Cornyn's bill, reintroduced his own gun safety proposal Tuesday.
In addition to the addressing lapses in the NICS system, Cruz's plan would criminalize "straw purchasing" of firearms for other people, expand gun rights for members of the armed forces, and target a controversial Obama-era program aimed at tracking firearms in the hands of Mexican drug cartels.
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