1. Literally, so close that one can or can nearly grasp it with one's hands. A possessive adjective or proper noun can be used between "within" and "grasp." I had to shimmy my chair over to the counter so the TV remote was within grasp. The ball was within his grasp as he dove into the air and stretched his arms out, but it slipped through his fingers at the last moment.
2. Very near to being obtainable or achievable. A possessive adjective or proper noun can be used between "within" and "grasp." The Super Bowl championship was within their grasp, when the quarterback slipped on the AstroTurf and fumbled the ball, allowing the other team to score a last-minute touchdown and win the game. This novel is a masterpiece, Sarah. I feel like the Booker Prize could be within grasp for you.
3. Very close to being apprehended. A possessive adjective or proper noun can be used between "within" and "grasp." Police have issued a statement that the fugitive is within their grasp, and that he should be in custody within 48 hours. The rogue agent was within grasp, but he managed to elude the CIA at the last moment.
*within one's grasp
1. Lit. where one can grasp something with one's hand. (*Typically: be ~;, get ~; get something ~.) The rope was within his grasp, but he was too weak to reach for it.
2. . Fig. [for something] to be obtainable; [for a goal] to be almost won. (Does not involve grabbing or grasping. *Typically: be ~; get ~; get something ~.) Victory is within our grasp, so we must keep playing the game to win. Her goal is within her grasp at last.