After being an unused substitute for England at Euro 2016 under Roy Hodgson, Barkley was left out of new manager Allardyce's first squad, for the World Cup qualifier in Slovakia last weekend.
That was despite the 22-year-old midfielder making a bright start to the season with his club which included him scoring two goals in his first three games in all competitions.
His fourth match, the 1-0 Premier League home win over Stoke in which he looked to have put in another creditable display, was watched from the stands by Allardyce, whose squad announced the next day was nonetheless missing Barkley's name.
When asked about that ahead of his side's league trip to Sunderland on Monday, Toffees boss Koeman said: "Let's talk about Everton, because we are now behind, after the international break.
"What he needs to do is to play good football, to show his qualities and show Big Sam every weekend that he needs to select the player.
"That is the best answer for the player by himself."
Barkley - holder of 22 senior England caps - has been just one of those to shine in what has been an encouraging opening made by Everton in their first term under Koeman's management.
Their three league games to date have yielded seven points, taking them up to fourth in the table.
Koeman missed out on at least one target on transfer deadline day in France midfielder Moussa Sissoko, who left Newcastle to join Tottenham instead, but the Dutchman did bring in Ecuador forward Enner Valencia on loan from West Ham
He is a potential debutant against Sunderland.
Valencia's arrival follows that in the preceding weeks of Yannick Bolasie, Ashley Williams, Idrissa Gueye and Maarten Stekelenburg.
And Koeman, who left Southampton to take charge at Everton in June, is pleased with how things have been shaping up so far during his tenure.
"I'm very happy with the squad," he said.
"Disappointed about our last target, but overall happy because the club did really a good job in the transfer window and the team is stronger than it was in my opinion.
"I am impressed by the whole spirit in the team.
"I think the new signings that we brought in showed a very fast adaptation to the team, to the rest of the players
Overall, it is very good."
Source : PA
Source: PA