With current skipper George Gregan likely to retire after the World Cup in September, speculation has been rife over who will take over.
While Connolly would not confirm Giteau as Gregan's successor, he predicted the 24-year-old would one day lead the side due to his professionalism on and off the field.
"I don't know if he will succeed George but I have no doubt he will captain Australia one day," Connolly told the Daily Telegraph.
"I've got to know Matt this year. He's very professional, he has such a strong presence within the team.
"On the outside, people may not see that, but Matt is a special player and he will have a big influence on Australian rugby."
Connolly continued to heap the praise on Giteau, comparing him to one of Australia's great centres.
"He reminds me very much of Tim Horan - just in his manner and how he acts," Connolly added. "He's such a strong personality.
"When he loses it hurts him badly.
"He's a bit cheeky but, like Tim, he's a winner and a leader of men."
Connolly even hinted that Giteau could take over at scrum-half for the rest of 2007 after performing admirably at number nine during the Wallabies' Spring Tour.
"George, like most guys, is probably under pressure," said Connolly.
"I had a chat to George last week about the Wallabies. What we do with Giteau is an issue.
"(Giteau) is now comfortable at nine or 12 (inside centre).
"Initially he was very nervous about playing half-back but that first 20 or 30 minutes against Wales (Giteau's starting Test debut at half-back) was probably the best footy that we've played this year."
Connolly added, however, that Gregan had been suffering from a shoulder injury that hindered his 2006, and was not keen to write off the 33-year-old just yet.
"A lot of people don't know George had a shoulder problem which affected the speed of his pass," he said.
"He also had a foot injury, he was a tough customer for us," he said. "I was happy with him but George, like everyone else, has things to work on.
"The team will be picked on form and George knows that.
"He said to me he has one goal - making the World Cup."