Updated at 11:58 a.m. ET

President Trump on Friday indicated that he supported new legislation on "intelligent" background checks for gun purchases after recent mass shootings in Texas and Ohio.

"On background checks, we have tremendous support for really common-sense, sensible, important background checks," Trump told reporters at the White House.

The president said the issue "isn't a question of NRA, Republican or Democrat," and added that he had spoken with the head of the National Rifle Association.

"We will see where NRA will be, but we need meaningful background checks," Trump said. "We don't want people who are mentally ill, people who are sick, having guns."

Trump touted his "great relationship" with the gun rights group, despite his apparent willingness to break from it on gun legislation.

"They supported me very early and that's been a great decision they made — we have Justice Kavanaugh, we have Justice Gorsuch, and they feel very strongly about the Second Amendment," he said, referring to the two justices he got confirmed to the Supreme Court.

Trump's remarks come one day after Senate Majority Leader Mitch McConnell, R-Ky., said the Senate would discuss various gun control measures when it returns to Washington in September.

The president told reporters that McConnell is "on board."

"I will tell you I spoke to Mitch McConnell yesterday," Trump said. "He's totally on board. He said, 'I've been waiting for your call.' "

In an interview with WHAS radio in Kentucky, McConnell said the Senate would likely debate background checks, "red flag" laws and assault weapons.

"What we can't do is fail to pass something," McConnell said, adding, "The urgency of this is not lost on any of us."

Democrats and Republicans in the Senate are likely to disagree about what proposals to focus on, but Trump told reporters he had a "great talk" with Senate Minority Leader Chuck Schumer, D-N.Y., and House Speaker Nancy Pelosi, D-Calif., about the issue.

"I think that the Republicans are going to be great and lead the charge, along with the Democrats," Trump said.

The House of Representatives passed a bill in February that would strengthen background checks on gun sales, but McConnell has so far refused to take that bill up for a vote in the Senate.

Asked what Trump would tell students who may be nervous about starting school after mass shootings, the president said:

"Go and really study hard, and some day you'll grow up and maybe be President of the United States or do something else that's fantastic. They have nothing to fear, they have nothing to worry about."

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