The 18-year-old forward impressed during a recent trial period at Upton Park, scoring in first-team friendlies against Millwall and Gillingham in November while also netting a double in an under-18 fixture with Aston Villa.
And the Premier League club have seen enough to sign the teenager on a permanent basis, with the deal set to be officially completed when the transfer window re-opens on Wednesday.
A statement on the club's official website read: "West Ham United are delighted to confirm the signing of Peterborough United youngster Jaanai Gordon, which will be completed on 1 January 2014."
Gordon played seven times for the Peterborough first team, making his debut for the League One outfit at the age of just 16 in September 2012.
The Northampton-born youngster made three substitute appearances for Darren Ferguson's side this season, with his most recent outing coming in their 2-1 Football League Trophy win over Brentford on October 8.
