The relaunched Govan club will discover at the end of the week whether they will be playing in the Irn-Bru First Division or Third Division next season.
Longmuir told Press Association Sport: "Friday is a very important day for Scottish football, I will let others judge just how important. I expect to have meetings with the SFA and the SPL this week. We hope to achieve an agreement which will be for the betterment of the game in Scotland."
The vast majority of Ibrox fans favour a fresh start in the bottom tier after last week's failed bid to remain in the Scottish Premier League.
However, the First Division looks the likely destination after Scottish Football Association chief executive Stewart Regan claimed that dire financial consequences meant he could not allow Ally McCoist's side to resume life in Division Three.
Longmuir expects to meet with Regan and Scottish Premier League chief executive Neil Doncaster over the next few days before the decision about Rangers newco is taken by the SPL board, on behalf of its members, who would then look forward to a package of reforms which are likely to include the re-introduction of play-offs and changes to the distribution of finances and votes.
Meanwhile, former Rangers striker John MacDonald is baffled by the whole situation and claims the Ibrox club have been subject to a "witch hunt".
MacDonald, who played for the Gers in the late 1970s and 1980s, told Press Association Sport: "Are there any rules or are they making them up as they go along? Any notions or talk of sporting integrity has gone right out of the window.
"Rangers really should start in the Third Division and most fans want that so there will be no comeback in years to come.
"But Stewart Regan has held a gun to the head of the SFL clubs by more or less saying they have to vote Rangers into the First Division. So it looks like the club will be in the First Division next season."