Arsenal head to Old Trafford to take on United on Sunday, leading the reigning English Premier League champions by eight points.
Ramsey and Mesut Ozil have been key to Arsenal's impressive start to the season, which has seen Arsene Wenger's men win eight of their 10 league matches.
But while Moyes confirmed Ozil was offered to United before the 2013/14 campaign – he maintained his decision to not chase the former Real Madrid playmaker was justified – the 50-year-old Scottish manager explained how close he had come to linking up with Ramsey earlier in his career.
"Yeah, I was very close to signing Ramsey at Everton," Moyes said.
"I had him in my house and Arsenal beat me to it.
"As regards Ozil, we weren't close to buying him but something was mooted (to us).
"We didn't need that position at the time. It was put to us and it was just something at the time we didn't need.
"We had Shinji Kagawa and Wayne (Rooney) and people of a similar ilk so it wasn't quite what we required at that time. We were looking at different positions."
While Arsenal and United's disparate starts to the new season have seen many pundits install the London club as genuine contenders and dismiss the reigning champions, veteran midfielder Ryan Giggs has insisted the Old Trafford outfit remain confident of victory on Sunday.
"They have certainly started the season well but over the years we have played against some very good Arsenal teams," Giggs said.
"You know what you're going to get, they're going to keep the ball, they have got plenty of quality and they are in good form.
"They had a great win in the week against Dortmund so we are going to be up against it but at Old Trafford we are always confident, no matter who we are playing."
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