Murray, who is not even officially on trial at Notts, played in Solihull Borough's 0-0 draw with Ilkeston Town on Saturday in an attempt to try and get himself back to match fitness.
And Murray's agent, Gino Culberston said: "He's absolutely straining at the leash and Saturday's game wasn't a game I particularly wanted him to play in. But Adam was determined to do it, has come through unscathed and has rejoined Notts County for training.
"There is a possibility Notts would like to sign Adam on a non-contract basis ahead of their game on Saturday which would give him the option of playing in another game.
"But we won't make a decision on Adam joining a club on a permanent basis until the end of the month when he's fully fit, although 10 clubs, two of them from the First Division, have already expressed an interest."