Ramsey held talks with the Toffees and United before eventually signing for Arsenal in 2008, with the Gunners paying around £5m to Cardiff City for the young Welshman.
Moyes has claimed that he held talks with Ramsey in his own home and nearly persuaded the midfielder to join him at Goodison Park.
He told the Manchester Evening News: "I was very close to signing Aaron at Everton. He was in my house but Arsenal beat me too it.
"He's played really well, he's improved. He maybe now sees himself as one of the really important players and that can help players.
"He's certainly not letting them down with the goals he's scoring."
Ramsey will come up against Moyes tomorrow afternoon when Arsenal face United at Old Trafford, fresh from scoring in successive wins over Liverpool and Borussia Dortmund in the last seven days.