The Villa boss been widely criticised by supporters after overseeing a poor run of form which has dragged his side into the relegation scrap.
But vice-captain Dunne, who had been out since February with a broken collarbone, has pleaded with fans to put aside their criticism and help the team.
“It's very important for the fans that they support us, because they want to be watching Premier League football next season,” he told the Daily Mail.
“We are giving our best and things just haven't gone right for us. But we've got three weeks to go and we need three weeks of massive support from everyone.
“We need everyone behind us and hopefully they will see the benefit of this season being a season when we've blooded six or seven youngsters in the first team and we can build on that and learn from this experience.”
|5/1||Aston Villa are 5/1 to be relegated this season with Paddy Power|
Former Rangers manager McLeish relegated Birmingham City last season before moving to Villa Park, but Dunne does not think the city rivalry is the reason for animosity toward the Scot.
“I don't think it's the fact he's come from Birmingham, it's the fact the results haven't gone our way and the fans have been frustrated themselves,” he added.
“We work with him all day, every week, and we know how much work he is putting into it. He is the person who keeps our group going and keeps us all motivated, so we don't like to see him getting any stick because it's us, at the end of the day, who are out on the pitch.
“I know the fans have their issues and stuff but we need them to get behind him, to get behind us, to get behind everyone.
“He is doing his best for us. He's been unfortunate with the injury list that he's had all through the season.
“It's hard, we just need the support now for a few weeks.”