There are 250+ million privately-owned firearms in the United States.4
4. BATFE estimated 215 million guns in 1999 (Crime Gun Trace Reports, 1999, National Report, Nov. 2000, p. ix , www.atf.gov/firearms/ycgii/1999/index.htm. The National Academy of Sciences estimated 258 million (National Research Council, Firearms and Violence: A Critical Review, National Academies Press, 2005).
And they've been running about 10 million "Form 4473"background checks for firearms transfers every year since then.
Of that 10 million, some are for transfers of "used" (already existing) firearms; some are for multiple purchases of new firearms. It's also my understanding that holders of Concealed Carry permits can buy without an additional background check.
The number of firearms in private hands is probably closer to 350 million.