There were fears that the 22-year-old would face a long spell on the sidelines. However, the positive diagnosis means the Glaswegian could be involved in this weekend's league encounter with Arsenal.
Martinez said: "The scan has shown up a sprain in his ankle and we need to see how that develops.
"Any sort of injury is disappointing, but we are relieved that it is not a really bad tear to his ligament. James has been working really hard this past two days, accelerating the recovery process.
"But it's too early to tell if he'll be okay for the weekend. It's always a worry when a player has to come off during a game.
"But, knowing James, his recovery period will be a lot shorter than most other players.
"He hasn't rejoined the group yet and he'll be doing his own training routine.
"It will probably be Friday before we find out whether he'll be all right to play."
Martinez added: "You knew it was a bad injury straight away because he got caught right on the ankle, and James' reaction was completely understandable.
"He knew he was hurt and he tried to stay on the pitch.
"He wanted to play the second half as well but the injury was too serious."