The X Factor boss and his partner are apparently set to name their son after his daddy, with Eric as a middle name – in honour of Simon's late father.
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It seems guests at Lauren's baby shower were aware of the choice – as some of them came bearing gifts bearing the initials "SC".
The American socialite celebrated her upcoming arrival with a blue-and-white themed bash held at Simon's Beverly Hills home in December.
Thirty-six-year-old Lauren is currently on a 'babymoon' – a pre-baby break taken by parents-to-be – with Simon in the Caribbean.
The couple are currently cruising aboard the Silverstream yacht in St Barts after spending some time in Barbados.
After enjoying some winter sun the couple will fly to New York, where Lauren will welcome her baby next month.
The expectant mum has chosen to have the baby in the city so she can be close to her son Adam.
After giving birth in one of the city's most exclusive hospitals, Lauren will take Simon Junior to the $26-000-a-month apartment they are renting in Manhattan.