The 53-year-old is expecting a baby with Silverman, the ex-wife of Cowell's former good friend Andrew Silverman, reported Sun Online.
According to reports, Cowell is considering honouring his dad by giving the tot the moniker Eric if it's a boy or Erica is it's a girl.
"Simon's made it clear he'd like Eric to be considered as a name. Lauren likes it as it's unusual in America, and quite stylish too. She also feels Erica's a cool name for a girl," a source said.
The X Factor boss father, Eric Philip Cowell, passed away after suffering a heart attack in 1999 at the age of 81.
Lauren, 37, is three months along in her pregnancy. She divorced from the father of her seven-year-old son, Adam, earlier this month when news of her affair with Cowell emerged.
The TV mogul has called his father a brilliant mentor in the past and recently spoke about his excitement of being a first-time parent.
"I'm proud to be a dad. Things are changing in my life right now. It's something I haven't thought of before and now I know I feel good about it," he had said.