City dropped into the second-tier European competition as their maiden Champions League campaign came to an end on Wednesday night. Goals from David Silva and Yaya Toure gave Roberto Mancini's men a comfortable 2-0 win over an under-strength Bayern Munich at the Etihad Stadium but Napoli's defeat of Villarreal left them third in Group A.
Barry said: "The players here are hungry to win trophies and if that is the Europa League then we are going to get up for it. We have got a squad here that can hopefully compete throughout the whole season in all competitions. The manager said afterwards he wants to win it so I am sure he will be putting teams out to win the games."
Despite that, Barry admitted there was no hiding from the disappointment of Champions League elimination.
City may have played with the verve that has helped them establish a five-point lead at the top of the Barclays Premier League on Wednesday night, but the damage was done earlier in the campaign.
Defeats away to Bayern and Napoli, and a home draw with the Italians, proved costly and City knew their hopes were slim prior to their final fixture.
Barry said: "Obviously, no matter how you go out you have still gone out.
"We got our job done (last night) but it is always tough mentally to put in a professional performance when it is out of your hands.
"At the end of the second half the atmosphere dropped. You could tell the fans were getting frustrated at a 2-0 performance - so you knew what was happening.
"This time it wasn't to be and we look back on a couple of games against Napoli that probably let us down in the end."