Manager Roberto Martinez, having taken over from David Moyes in the summer, lured the ex-England player to Goodison Park from Manchester City with the prospect of regular football.
Barry is out of contract at the end of the campaign and Ratcliffe believes the 33-year-old's form means other clubs may be preparing to make a move.
Speaking on behalf of official FA Cup Supporter williamhill.com, he told Press Association Sport: "No doubt about it, Barry is the best piece of business Roberto has done.
"Unfortunately Gareth is the wrong side of 30 but he looks fit enough to go on and it would be nice to think Roberto can sign him next season b ut the kind of performances he has put in this season (means) other clubs will be in for him.
"He wouldn't go amiss at Arsenal. He is that good a player you would think they would want that type of player."
Ratcliffe hopes a deal can be done for Barry, especially as he does not believe Everton have the cash to afford £20million-rated on-loan Chelsea striker Romelu Lukaku.
"If a player is out of contract, the money he is looking for should be acceptable because you are not paying a fee for him," added the former centre-back, who 30 years ago lifted the FA Cup with Everton.
"That means there should be some leeway in which Everton can compete with the bigger payers.
"But I don't think Everton will be able to afford £20million for a player unless they are looking at selling someone and when you are talking figures like that you are looking at Ross Barkley.
"We do need a striker and while there are still things missing in Lukaku's game he is built for the Premier League."
Martinez's arrival in the summer has heralded a more open, attacking style at Goodison and the players appear to have been energised by being given a freer role.
Ratcliffe likened the Spaniard to his great mentor Howard Kendall, who won two league titles, the FA Cup and European Cup-Winners' Cup with the Toffees.
"We had Howard Kendall come in and all of a sudden we were boys who were treated like adults," he explained.
"He threw a responsibility towards us and we reacted and that is what Roberto has done.
"The game is a little bit different and I don't know what Roberto is like off the field but he has given players responsibilities and they have responded to that really well.
"I think he has given them a non-fear factor. Under David Moyes I don't know whether they were fearful - fearful maybe of a Dave backlash after the game.
"Roberto wants you to have balls under pressure and not to be afraid to lose it and I think that is the difference.
"He is encouraging you to get the ball again rather than being afraid of making a mistake."
:: williamhill.com are an Official Supporter of the FA Cup. Latest oddds: 4/5 Arsenal, 13/5 Draw, 17/5 Everton.
- Sports & Recreation
- David Moyes