The two midfielders started together in the draw with Tottenham Hotspur.
The Blues kicked off their Premier League campaign on Saturday as they took on Tottenham Hotspur at Goodison Park.
They impressed too with an impressive display seeing them secure a draw in their opening game.
Everton's Idrissa Gueye in action with Tottenham's Dele Alli
Gueye in particular was in fine form, earning rave reviews from the Everton faithful, who were quick to liken him to French star N’Golo Kante.
Barry was also a strong performer alongside him, and Koeman has hinted that they will be his central midfield pairing this season.
“Idrissa is the kind of player that puts in a lot of energy, he’s aggressive and he can also play good football. He will make a good tandem with Gareth Barry,” he told the Times.
Tottenham's Kyle Walker in action with Everton's Gareth Barry
It now remains to be seen if Gueye can continue to impress for Everton after this strong start to life at Goodison Park.
The comparisons to Kante were not over top in any regard; with him showing the same energy and defensive know how that made the Frenchman a star last season.
If he can continue in that vein then there is no denying that the Toffees may have secured themselves one of the signings of the summer.
Everton manager Ronald Koeman applauds fans after the game