Man Utd's Rose in Bristol City trials
He did not feature for United's first team but spent the second half of last season on loan at Oxford, making 20 Conference appearances for the U's.
"I've seen enough since I've been here to know I'd like to sign a contract here," he told Rovers' website.
"The mentality of the club is perfect for me. I managed to score against Cowdenbeath on Thursday, though I'd like to get more games and minutes under my belt."