Arsene Wenger is out of contract next month and he is yet to decide whether he'll sign a new deal.
The French boss has confirmed he will meet with the Arsenal board next week to discuss his future before the club make an official announcement.
Wenger guided Arsenal to a record 13th FA Cup with a 2-1 victory over Chelsea this evening and Ramsey is hoping the 67-year-old signs a new deal.
"Of course I want him to stay at club," he said.
"He deserves this. We changed the formation and had a lot of success. Fair play to him, he changed the system. Hopefully, he'll be there next season."
Arsene Wenger celebrates with Aaron Ramsey after Arsenal's FA Cup triumph
Alexis Sanchez opened the scoring for Arsenal at Wembley after just four minutes.
Chelsea responded in the second half as Diego Costa equalised before Ramsey netted the winner with a brilliant header 11 minutes from time.
"I can't describe this," Ramsey added.
"It's been an up and down season. We've finished with the FA Cup so it has to be successful.
Arsene Wenger lifts FA Cup as Arsenal players celebrate Sat, May 27, 2017
Arsene Wenger was the centre of attention during the wild Arsenal celebrations
Play slideshow Getty Images 1 of 11
Arsenal celebrate winning the FA Cup against Chelsea
Get Quotes on Home Insurance
Aaron Ramsey wants Arsene Wenger to stay at Arsenal
"I've got the winner again. I love this competition. The boys deserve this. I'm happy for the manager. To get another FA Cup – delighted."
Danny Welbeck added: "It was a great team performance.
"We stuck in there when it looks like things were against us. You always worry when you are a goal down and it was a game full of surprises.
"The manager is his own man and he makes his own decision and the board will make the right decision so I can't comment on that."