McCoist's side clinched the Scottish League One title in midweek following a 3-0 win over Airdrie before businessman King flew into Scotland on Thursday night ahead of meetings with Light Blues' supporters and investors.
The former director will also head for a showdown with club chairman David Somers after advising fans to withhold their season-ticket cash as concerns linger over the state of the finances.
The ruling Rangers regime - which presided over last year's £14.4million loss - have accused King of "damaging and destabilising" the club with his statements.
But McCoist hopes the two sides can now find a way to end the battle for boardroom control.
He said: "I won't be privy to any of the meetings but I just hope that we can all move forward afterwards.
"Like any Rangers fan, I would be very hopeful the outcome would be positive and then we will see what happens after that."
McCoist again refused to discuss the specifics of the rumbling row but did admit it had been a distraction to his job of managing the team.
"Both the board and Dave have said it is best that I concentrate on the football," said the club's all-time record scorer.
"A few months ago, it was perhaps part of my job to get involved (with the off-field matters). But now that there is a proper board in place and Graham Wallace there as chief executive, that side of things will be left to them.
"I'm not sure if that makes my job any easier but it certainly allows me to focus 100 per cent on the football, which in all honesty I haven't been able to do in the past. If that is a criticism of myself then so be it.
"You hope things would calm down and they have calmed down a little bit. But obviously there are still ongoing issues and I appreciate that.
"But I have to say I think the club is in a far better place than it was this time last year. Now we just need continued improvement on and off the park and that's what we hope to do."
Victory over Airdrieonians gave the Gers an unassailable 26-point lead over Saturday's opponents Dunfermline but McCoist knows tougher challenges await his side next term.
"We have got the job done comfortably with eight games to go," he said. "We've achieved our target for this year but it will be very difficult next year.
"But nights like Wednesday mean everything to the people at the club and the supporters.
"To have well over 40,000 there on a Wednesday night at the same time as Barcelona are on the television is an unbelievable crowd.
"The title is payback for the fans. We want to give them what they want - a winning team."