Rooney is believed to be open to a move to the Emirates Stadium as long as Arsene Wenger's side are able to match his salary demands.
Arsenal's second-biggest shareholder, Alisher Usmanov, has also praised the 27-year-old and claimed that it would be a "great success" if the north London club signed the England international this summer.
But Adams, who spent 22 years at Arsenal, told talkSPORT: "I'm not sure Wayne Rooney could handle London, to be completely honest. That would be a massive risk, for me.
"I don't think it's a question of finances. I think it's his temperament and mentality. London is a different animal. I think it takes a different type of player [to handle it]. think Wayne is unquestionably a super player and I'd love to see him at the Arsenal. Technically he's a fantastic player but I'm not sure about him mentally, off the pitch and his professionalism.
"I know Sir Alex Ferguson has dealt with Wayne brilliantly over the last few years and I'm sure Arsene [Wenger] could handle him but I'm not sure. I wouldn't like to spend my money on Wayne."