Jon Stevens, who replaced late singer Michael Hutchence as frontman for INXS before joining Noiseworks, allegedly hit Eldar in a collision in the tony Sydney suburb of Point Piper on Monday, the Sydney Morning Herald reported.
According to the report, Eldar is in Australia visiting family, and was en route to visit his elderly mother-in-law when the incident occurred.
The Israeli journalist underwent surgery on his right arm and left leg and is expected to have further operations in the coming days, the report said.
Eldar's sister-in-law, fashion designer Charlie Brown, called the incident "unfortunate for all parties", the paper said.
Yahoo reported that Stevens got out of his car to offer assistance to Eldar after hitting him. The musician was questioned by police and then released, pending further inquiries, the report said.
Stevens told Australia's Channel Nine on Monday: "I hope the gentleman is OK. Police and paramedics have assured me he is but I'll be checking in to make sure. It was just an accident. I can't take a trick right now."
Eldar was the award-winning political correspondent for Haaretz, where he worked for 35 years, before joining Al-Monitor. He is also an author, writing a biography of Shimon Peres and an in-depth look at Israel's settlement policies.