The US, co-hosts of the World Cup alongside Canada and Mexico, will play Paraguay, Australia and a playoff winner in Group D.
Mauricio Pochettino's US side have beaten both Paraguay and Australia in recent months, making them favourites to progress from the group.
Australia coach Popovic, however, struck an optimistic tone when discussing his side's chances immediately after the draw, saying: "I am sure the boys are looking at the draw and can't wait to get over there."
In an appearance on a FOX Soccer podcast alongside former US internationals Cobi Jones and Brad Guzan, Donovan took issue with Popovic's reaction.
"I watched the Aussie coach's interview after the draw, and it was really interesting. It was unique," he said.
"Because, most coaches say all the same stuff you're expecting, like 'Oh, it's a difficult draw, this is going to be so hard', and his first comment was 'We're satisfied'. And I was like: 'Really? Are you?' He just had this smugness about him.
"And I'm just - alright, I can't wait. I wish I was on the field. I want to play these guys... They're a solid team, they're hard to play against, this is not going to be some blowout. But it's a team I think we should beat."
In their predictions for the group, Donovan, Jones and Guzan all tipped the US to sail through into the knockout stages.
"We all picked how we thought this would go and we're all aligned," said Donovan, who is the joint-record goalscorer for the US men's team.
"We think the US top the group. Turkey again, we're maybe getting ahead of ourselves, but we think that will be them (playoff winners), and we think they finish second.
"Paraguay (in third) and then thanks for coming Aussies and your smug coach. You can get back on the Qantas airplane and head back home, pal."
The World Cup will be held from June 11 to July 19.