Gordon, 18, impressed during a recent trial with the Hammers, scoring in the Under-18s’ victory against Aston Villa and two first-team friendlies.
He has played seven senior games for Posh since September 2012.
The Hammers have also been linked with a loan bid for Ghana’s Asamoah Gyan during the January transfer window.
Manager Sam Allardyce’s side are short of attacking options because of Andy Carroll’s long-term injury-enforced absence.
The club are 19th in the table after winning just one of their last 10 Premier League fixtures.
