The baby was born at 5:45 p.m. EST Friday at Lenox Hill Hospital in New York, TMZ reported.
The child is 54-year-old Cowell's first and Lauren Silverman's second.
She has an 8-year-old son with her ex, Andrew Silverman, TMZ said.
"He's incredible!" Cowell told the New York Daily News. "His name is Eric, after my father."
Simon Cowell said Silverman is doing "fine" and is "healthy." The baby weighed 6 pounds, 7 ounces, at birth, the new dad said.
"It feels great!" Cowell said of being a father.