Club directors met on Saturday after claims by former owner Craig Whyte that he was involved with Green's consortium when they purchased the business and assets of the club last summer.
Whyte's allegations were accompanied by taped recordings of conversations between himself, Green and commercial director Imran Ahmad.
The board has confirmed it will probe the claims of the former owner, whose non-payment of tax forced administration and subsequent liquidation.
Having endured months of turmoil last year, McCulloch is keen to avoid another summer of uncertainty.
The skipper said: "A bit of clarity would be great. And just the truth. I think everybody deserves to know the truth."
On the pitch, with the Division Three title wrapped up, McCulloch and Kyle Hutton were on the scoresheet as Rangers ran out comfortable winners in a 2-0 victory over Clyde.
"I felt we deserved the win," he said. "We could have had another couple of goals.
"We had a couple of half-chances and shots from distance but we dominated possession and we were deserved winners."
McCulloch's strike was his 25th goal of the season.
But he claims he will have no complaints if Andy Little - also on 25 - beats him in the scoring charts come the end of the season. He said: "If I don't get top goal scorer, it's not the end of the world for me.
"Being honest, as captain, I would rather Andy Little got it. As captain, sometimes you've got to put the team first and put individuals first.
"I know how much Andy wants it and it would be good for him to get it."