HGGC has 30 days to negotiate the finer points of the deal according to Reuters, which cited four people with knowledge of the matter.
Silver Lake bought Serena for $1.2bn in 2006, a value it is thought it will struggle to equal in a sale, with the deal expected to value the company at between $450m and $500m.
It emerged in November that Silver Lake could sell Serena to private equity after talks with Attachmate Corp over a purchase reportedly fell through.
Silver Lake hired Barclays to run an auction process in the summer following at least one previous failed attempt to exit the business.
Attachmate was thought to have a strong chance of picking up the asset in a trade sale but ended talks before a deal could be reached.
Veritas Capital Fund Management and Platinum Equity were also thought to be in talks to buy the business, which provides software to help businesses streamline their management processes through their IT infrastructure.
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