Clark damaged knee ligaments after catching his studs in the turf during training but Dunne is the major concern after it was confirmed the centre-back had broken his shoulder in the dying stages of Sunday's 1-0 defeat at home to Manchester City.
Dunne suffered the injury in after falling heavily when challenging England keeper Joe Hart for a cross and will undergo surgery in Birmingham later on Monday.
A club statement confirmed: "Richard Dunne sustained a fracture to his right clavicle in the game against Manchester City. He will undergo an operation today in Birmingham. It is anticipated that he will be sidelined for up to eight weeks.''
Manager McLeish, whose Villa side are on a run of just two wins in their last nine Premier League games, said: "As for Ciaran, he's tweaked a ligament and we will lose him for a few weeks. It's not an overnight return either.''
Without a home win since November 5 - a run of five defeats and two draws - these are worrying times for Villa, who are just seven points above the relegation zone.
Following Sunday's defeat, a clearly unhappy McLeish said: "Our home record is pretty abysmal. I'm disappointed with that because in my managerial career I pride myself with home performances, and the current one isn't good enough.
"No club under 40 points can afford to be complacent in this league. Everybody is going to have a right go, and it's going to be a big fight.''
It could prove too much for some fans, up to 100 of whom voiced their concerns before kick off with a protest against McLeish.
There were a few banners and a few chants denouncing the Scot, but they were a small minority as the vast majority are preferring to give him a chance and get behind the team in its hour of need.
"I thought the fans inside the ground were brilliant,'' McLeish said. "I said to the players in the team talk that the fans want them to sweat blood when they are wearing the jersey, and we have to show we are prepared to do that.''