There has been much speculation over the Dutchman's future ever since he announced he would not be signing a new deal with the Gunners, and criticised the club over their transfer policy and the direction in which they were headed.
The 28-year-old scored 30 Premier League goals last season which went a long way towards helping Arsene Wenger's side secure third place, and midfielder Ramsey, currently preparing for the Olympics with Team GB, knows how difficult it would be to replace those goals.
He told the The Belfast Telegraph: "We don't know anything that goes on. We hope that he stays at Arsenal.
"He had a brilliant season last year and he is a big player for us, so we are just hopeful that he will stay."Arsenal have already signed forwards Lukas Podolski and Olivier Giroud, perhaps with Van Persie's imminent departure in mind.
Arsenal's poor start last season was attributed, by many, to Wenger leaving their dealings in the transfer market until the last minute.
Ramsey is convinced if they get their business done early they can make a better start to the upcoming campaign and possibly close the 19-point gap to second placed Manchester United.
He added: "If we get off to a better start this season than we did last season, we know we are capable of finishing pretty strongly, like we did last year.
"We have brought in a couple of players who are well-known and hopefully we can put ourselves in a position to compete early on."