The Czech Republic international has largely been out of the side this season, but netted a brace on Saturday to secure the Gunners a valuable 2-1 win against West Bromwich Albion at The Hawthorns.
The 32-year-old has been linked with a return to Borussia Dortmund after an injury-ravaged career at the Emirates Stadium, but Wenger is keen for Rosicky to end his career in North London.
"I thought he was fantastic - he's not a player I want to lose," said Wenger. "I rate him very highly.
"He has one year left on his contract but I hope he stays and finishes his career with us here.
"He's had a difficult season and been injured. But he always seems to come in and gives us a lift at the end of the season. He's a quality player."
Rosicky joined the club in the summer of 2006.