The former United defender was brought in as part of Moyes's coaching team when the Scot succeeded Sir Alex Ferguson.
However, when Moyes was sacked by the club in April, Neville remained at the club and worked under interim manager Ryan Giggs. Keane went on to hint that if he were in the same position, he would have left the club.
"David Moyes brought in Phil Neville and everybody said it is great to have Phil Neville," Keane told the Daily Star. "I would like to ask, 'What did Phil bring to the party?'"
"And when David Moyes left, why didn't Phil leave with him? Because David brought him in with him. So, interesting.
"Should Phil have walked with him? I don't know, you would have to ask Phil but I know what I would have done. But we are all different."
Neville, along with goalkeeping coach Chris Woods, have reportedly been told that they will be leaving the club.