When he first joined he was thrown into a couple of games and in some of those it was because we didn't have any other option, if memory serves this was the case against Arsenal in the League Cup. Since that handful of games in the early months of this year, he's had some injury problems again and Conte has constantly said throughout the year that he needs time, that he's not ready and that we need patience with him. The press asked about him a few times and Conte said the same thing every time.
We still have to see what Sarri's plans are but perhaps there's space for him in the 4-3-3 (if he goes with what he did with Napoli) with Kante and Jorginho. Of course, the first thing it depends on is whether he can stay injury free and fit, then if that's the case, I don't think he's expected to be a first teamer but rather a regular squad player. The most important thing with Barkley's future though is Sarri, the tactics and whether or not he takes a liking to him. I would say though that the future under Sarri doesn't look bright for the three Englishmen, Cahill, Drinkwater and Barkley but Barkley probably has the biggest chance to have a bigger role out of the three if he stays fit.