The Bermuda international, 20, has been on trial with the National League North club throughout March.
Last week he was on Concacaf Cup duty with his country in the Dominican Republic, winning his eighth cap.
Manager Tommy Wright said: “Osagi has trained with us for a couple of weeks, and has impressed enough to earn a deal until the end of the season.
"We would like to welcome Osagi to the club, and wish him all the best with us."
Darlo are looking to end the season strongly and on Saturday ended their five game winless run with victory at bottom club Nuneaton.
The Quakers are next in action at home to Chester on Wednesday - and Bascome could feature.