The star, who placed second in the show in 2009, is in talks to appear on the panel, E! News reports.
"They have been talking about Adam for over a month," a source said. "He personifies the show, and will be a popular choice right from the start."
The insider continued: "He knows what it's like to be on the receiving end of the judges' comments, he has a human touch, and they know they can't go wrong if they bring him on. He would be able to bring a unique perspective to the show. People love him."
Meanwhile, Lambert told Digital Spy: "If I was asked to do it, I would be thrilled to."
Lambert is currently promoting his new album Trespassing and recently performed on stage with Queen.
Rumours have been circulating about Jennifer Lopez's future on American Idol since the latest season ended in May, with the judge herself admitting that she is not sure whether or not she will return.
Speculation is also building around the future of current American Idol judges Randy Jackson and Steven Tyler, with a source telling E! News: "Randy is great, but he's part of the old Idol."
Elsewhere, another insider claimed that Tyler's future is "all up in the air".
A Fox executive recently claimed that he believes American Idol could continue up to 17 seasons.