Keane was named in Martin O’Neill’s final 23-man squad for the tournament in France, despite sustaining a calf strain during training ahead of Ireland’s final warm-up friendly against Belarus.
The LA Galaxy striker did not feature in the 1-1 draw with the Netherlands last Friday and had been due to start against Belarus at Turner’s Cross before injury scuppered O’Neill’s plans to give the 35-year-old a game prior to the tournament.
“[Keane] thinks that he could be [fit for the first game]. If it’s not Sweden, he thinks he’ll be fit for the second game,” O’Neill told reporters after announcing his squad for Euro 2016.
“That’s the thought at this minute, but I am not a doctor.”
Like Keane, McCarthy did not play in either friendly, but has been training separately from the rest of the squad with a fitness coach and completed a session on Tuesday, giving O’Neill a sense of optimism about the Everton midfielder’s chances of playing in France.
The Ireland boss added: “James McCarthy did a little bit of work [on Tuesday], it’s a good improvement in that sense and that helps a lot.
“I’m not a soothsayer, but I think there’s a fairly decent chance that they’re going to be ok.”
Ireland kick off their European Championship against Sweden in Paris on June 13.
