Law enforcement sources across the country tell TMZ ... officers aren't blinking at a record number of Americans -- 3,740,688 according to the FBI -- applying for background checks last month alone.

Cops in L.A. tell us, if people knew how many criminals are out there armed, they would not be worried about this current surge. If anything, cops tell us the biggest concern is whether all these new owners have safe places to store their weapons ... such as safes, gun lockers or lockboxes.

Meanwhile, Las Vegas Metro PD sources tell us they have no problem with people buying guns, as long as they're legal. They also hope new gun owners have had or will commit to proper training.

Cops in Miami echo that ... saying the rush of new sales is all good because they're getting background checks, instead of buying illegal guns off the street.

As we told you ... guns and ammo were flying off the shelves in Florida just hours after Gov. Ron DeSantis implemented a "stay at home" order. And remember that nearly 4 million figure from the FBI? It's the first time it's ever gone over 3 mil in a month since 1998 ... when the FBI started recording sales.

At this rate, the FBI and the National Instant Criminal Background Check System will perform more than 36 million background checks in 2020 ... easily beating last year's record of 28M. This, of course, doesn't mean everyone that applies for a background check will actually buy a gun ... but the numbers are startling, for sure.