The online bookmaker cut the price from 40/1 to 1/3 as they saw a sudden run of stakes on the Manchester City midfielder moving to Goodison Park, having previously seen little money in the market.
Barry, who made 27 starts for City last season, is facing increased competition for a starting berth following the arrival of £30million signing Fernandinho.
The 32-year-old had featured throughout City's pre-season preparations but missed their final friendly, a 3-1 defeat to Arsenal.
Sky Bet Football Trader Chris Spicer said: "It struck us as fairly strange to be seeing such a run of bets on a market which had previously captured little interest.
"However, with Marouane Fellaini odds-on to leave Everton for Manchester United, Barry could be seen as short-term cover, so a loan move certainly makes some sense, especially given the competition for places in the middle of the park at City."