The defender has missed the last two matches with an ankle injury he picked up in a training session on the Emirates Stadium pitch after the win over Leeds.
Hincapie will not be involved when the Premier League champions take on Bayern Munich on Wednesday night, though he is back on the grass doing individual training.
Mikel Arteta had initially suggested that the 24-year-old would be on the sidelines for "weeks", but Hincapie is now pushing to return when the Gunners host Everton on Saturday.
Close: Piero Hincapie
Getty
"Piero, let's see," the head coach said.
"If he is able to train the next few days he can be available for Everton, which is great news."
Arteta also confirmed that Martin Zubimendi is available and "will be involved" against Bayern after overcoming his own ankle issue.
Arsenal are in Munich for their toughest test of the season so far as they look to make it 11 wins from 12 matches in all competitions.
Despite that incredibly strong start to the campaign, debates have once again picked up over the team's style of play.
Boosted: Mikel Arteta
PA
That was also the sixth game in a row in the Premier League and Champions League in which Arsenal have failed to score in the first half.
However, when asked if he was aware over discussions on how Arsenal play and whether more is needed in attack, Arteta said: "I don't have that impression.
"I heard that the style was incredible because we have scored so many goals, we have more possession, more shots, different players participating.
"No goals from set-pieces [this season] so the story is completely different to last year.
"It will be a part. I think we are talking about a very small minority. The other ones, I think they have enjoyed a lot the way we have played and dominated games this year.
"I think we have proven that. That's my opinion."