The Wales international has failed to agree a renewal of his contract, which expires at the end of the season, and he could be set to depart in January.
Ramsey is regarded as an influential figure with the Gunners, having enjoyed 10 years at the club, scoring 59 times in 338 appearances across all competitions.
Wilshere, who also allowed his contract to run down at Arsenal before joining West Ham this summer, feels his former club are foolish to let Ramsey go.
He believes Ramsey has all the attributes to become a future captain for the north Londoners and has urged them to open up contract talks before it's too late.
Wilshere told the Islington Gazette: "He's 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."