"I went to see a scan today and it is just surreal," he told The Times newspaper. "You see this thing which is now alive moving around. So I am happy."
The TV talent show judge is looking forward to welcoming the new arrival, due in 2014. He had previously ruled out becoming a parent, but now says he feels overcome by prospect of fatherhood.
"The fact that it can happen at this time of my life is fantastic, maybe a miracle," the 53-year-old told Mail Online.
"Once I heard the news about the baby... little bit of a shock, then I got used to it, and then I became incredibly paternal and protective and I feel very attached to the baby already."
Lauren is thought to be 12 weeks pregnant with Simon's baby. She reached a divorce settlement with her husband of ten years last month.
Her property mogul husband filed for divorce shortly after the news broke of his wife's infidelity with his close friend.
They have also come to an agreement to share joint custody of their seven-year-old son.
Simon - whose fortune is estimated at $368million - was cited as a "co-respondent" in the court papers for adultery.