The 18-year-old has impressed since breaking into the Arsenal first team and scored his first senior goal for the club in the 2-1 win at Swansea last month.
That came a week after he made his full Barclays Premier League debut in the win over Stoke.
Rosicky, the Czech Republic forward, has been impressed with Gnabry's contribution and anticipates further progress under Arsene Wenger. "He has a manager here who will develop him and who has a history of developing young players," said Rosicky. "Nothing better could happen to him than being offered a new contract here.
"He is a good dribbler and great in the one-on-one situations. That can be his strong point in the future and for an 18-year-old he is doing very well. He is a very talented boy and will find no better place to learn than here. I think it works very well for both sides that he stays here."
Gnabry, who signed his first professional contract last year after shining in the club's youth ranks, made his Arsenal debut in the 6-1 win over Coventry in last season's Capital One Cup.