Ramsey's contract with the Gunners is poised to run out at the end of the season, and it's been reported that Arsenal withdrew their offer of an extension, setting up either a January exit or a move at the end of the campaign.
Wilshere, who is now at West Ham United, spoke to the Islington Gazette and said he thinks Arsenal should be looking to build a team around Ramsey instead of letting him leave (h/t Chris Burton of Goal):
"[Ramsey has] grown up to be a great player with Arsenal. They could do with someone like him. We know what he can do for the club.
"For me he's a future Arsenal captain. I know he loves the club. It's difficult for me to comment on because I'm not there but I do know he's a great player—if it was me I know I would want to keep him around."
There have been a number of sides linked with moves for Ramsey.
James Olley of the Evening Standard reported that both Liverpool and Manchester United are admirers of the midfielder, while a trio of Serie A sides—AC Milan, Inter Milan and Juventus—are also said to be keeping an eye on the situation.
If Arsenal were to sell Ramsey in January, they would at least be able to bring in some funds for a player who has been with the club since 2008.
However, the 27-year-old has said he would like to stay at Arsenal until the end of the season, per Sky Sports Premier League:
When Wilshere and Ramsey were youngsters, Arsenal supporters would have envisaged the pair would be the club's first-choice central-midfield pairing in their peak years. However, the former left upon the expiration of his contract last summer, and the latter looks set to follow suit.
Ramsey has been a crucial player for Arsenal in recent years, fulfilling numerous roles in midfield and troubling opposition defences with his bursts into the penalty area.
He has been one of the team's most consistent goalscorers throughout his time at the Emirates Stadium, and he was productive again last term:
He has already laid on three assists in the Premier League this season, and he has scored one goal. That came against Fulham, as Ramsey started and finished a brilliant team move, helping the Gunners record a 5-1 win—their ninth victory in a row.
There doesn't appear to be any danger of Ramsey taking his foot off the gas in what may be his final months as an Arsenal player. Still, the prospect of him captaining the side this season and beyond feels unlikely.