According to Sky Sports, the transfer is worth an initial £45m and could rise to £50m with add-ons.
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Delap is expected to undergo his medical on Wednesday before completing the move.
The 23-year-old attracted interest from several Premier League clubs, including Everton, but Forest have won the race for his signature.
Chelsea signed Delap from Ipswich for £30m last summer and are set to make a significant profit on the forward. He scored two goals in 41 appearances last season.
Delap was left out of Chelsea’s squad for their Premier League meeting with Fulham on Tuesday, paving the way for his move to the City Ground.