In 2004, Paolo Pellegrini was out of a job, living in a one-bedroom apartment in Westchester County, N.Y., with little money in the bank. He soon managed to land a gig working for hedge-fund manager John Paulson.

Today, after helping Paulson & Co. score $20 billion with a bet against housing and pocketing about $175 million for himself, Mr. Pellegrini is an unnamed but key character in the government's lawsuit against Goldman Sachs Group Inc., according to people familiar with the matter.

The...