The 20-year-old joined Gillingham on loan from the Hammers last month.
Lee impressed during Saturday’s 3-0 victory at Cheltenham.
Gillingham boss Andy Hessenthaler told the club’s official website: “We saw the best bits of Olly and to be honest we haven’t seen that yet, we know what he can do.
“I’ve seen him at West Ham and I think he produced what we have all been waiting for and he will tell you that himself.”
