The 22-year-old, who won his first cap for Scotland in the narrow defeat to Ireland in Dublin earlier this month, has been on loan from Edinburgh Rugby since March.
He was part of the Glasgow squad that defended their Melrose Sevens title.
Head coach Gregor Townsend said: "Hugh (Blake) will add to the depth in the squad and has the ability to play anywhere in the back-row.
"He is an intelligent rugby player who has a big future in the game.