Influential board members are upset with McKenzie as the Wallabies, and the code, have suffered for the sad state of affairs surrounding Kurtley Beale's misconduct allegations.
But Moore cautioned against appointing a second new coach in 15 months - after Robbie Deans held the post for six seasons.
"When Queensland was going well everyone was saying Ewen should take Robbie's job, and when Jake was going well [with the Brumbies] it was Jake should take Robbie's job and now Link [McKenzie] is on the receiving end as well," the hooker said.
"These things go in swings and roundabouts but I think now Ewen should get all the support he can get."
Moore's comments come as Wallabies vice-captain James Slipper said players felt responsible for the results pressure McKenzie was under.
After going unbeaten for eight tests, leading into their last Bledisloe clash, the coach's record now stands at 52 per cent.
Moore met with leading ARU officials Ben Whitaker and Phil Thomson this week and is taking recommendations from players to help frame the "optimum" management structure following the departure of business manager Di Patston.
"It's to help Link now to establish a framework around him that's going to help us having success, not only beat the All Blacks this weekend but consistent success," he said.
While current skipper Michael Hooper has voiced strong support for Beale, whose future is in grave doubt for distributing lewd messages about Patston, Moore spoke out for the former staffer - who has also been subjected to questions about her relationship with McKenzie.
"What Di has had to go through is very disappointing," he said.
"No one should have to go through that ... and as a player I feel very disappointed that that's happened."
- AAP