The often quoted 300 or 400 million private guns in the US is very outdated. This FBI chart shows the number of background checks done, by month and year only since 1999. (another 498 million)
Most of the people that are not going to pass the check already know that, and it is a crime to buy a gun when you can’t pass the check. And the failure is reported to the local police. So most of these checks resulted in sales. We don’t track gun sales actually, but the NICS check is a reasonable number to use.
What I am leading up to is this:
NICS Firearm Checks: Month/Year — FBI
Between 2 and 3 million background checks each month.