President Trump insisted Tuesday the U.S. already has "very, very strong background checks" for gun purchases, noting many of his supporters "are strong believers in the Second Amendment," marking another sign he is backing away from supporting expanded checks.

In the wake of the mass shootings in Texas and Ohio, the president had said he was looking to implement "very meaningful background checks," claiming Senate Majority Leader Mitch McConnell is on board. But the president, who has spoken with National Rifle Association chief executive Wayne LaPierre several times in the last couple of weeks, most recently Tuesday, has appeared to take a more tentative position on gun reforms this week.

Under current federal law, guns purchased from a licensed firearms dealer require the buyer to pass a background check, but private gun sales between individuals carry no such requirement. Congressional efforts to expand background checks to cover all sales have failed several times.

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Mr. Trump said he isn't offering seemingly conflicting positions on gun control "to be cute."

In a meeting with Romania's president Tuesday, Mr. Trump said that while the current system is "sort of missing areas and areas that don't complete the whole circle," the system is overall "very, very strong." He went on to express fears about the potential for a "slippery slope" where "all of a sudden everything gets taken away."

"We are very strong on our Second Amendment — the Democrats are not strong at all on the Second Amendment. I would say they're weak on the Second Amendment," the president said Tuesday.

Mr. Trump said his team is "looking at different things" to stem gun violence.

"We're looking at mental institutions, which we used to have," he said. "Like, as an example, where I come from in New York, they closed up almost all of their mental institutions — or many of them — and those people just went onto the streets. And they did it for budgetary reasons."

Some of the staunchest gun rights groups have warned the president against expanding background checks or implementing "red flag" laws, extreme risk protection orders that allow authorities to confiscate the weapons of someone deemed to be a risk to themselves or others.

Kathryn Watson contributed to this report.