McLeish spent four and a half years at the Ibrox helm and won all three domestic trophies before parting company with the club in 2006.
He says showdown talks with then chairman Sir David Murray over funds for players proved to be the beginning of the end of his spell in Glasgow.
Speaking in an interview to be shown on BBC ALBA on Saturday night, McLeish said: "At the end of my Rangers career, David and I had a man-to-man chat about it and I said something has to happen, either you give me funds and I continue or I go and the players get a new face, they probably need a change - something needs to change.
"And because there wasn't the embarrassment of riches that perhaps there was in the past, we shook hands and I agreed to go, and it is a new chapter."
McLeish was succeeded by Paul Le Guen, who infamously lasted just 200 days before being replaced by Walter Smith.
Despite winning the SPL last season and securing Champions League football, Smith's only summer signing was Jerome Rothen on a season-long loan from Paris Saint-Germain.
Speaking when quizzed about funds being made available to him in January, the Rangers boss said: "I don't want to get into that kind of aspect of the club at the present moment.
"I would rather just concentrate on the fact that we have to make sure what we've got at the present moment manages to get itself together enough to keep us going and concentrate on getting a better level of result.
"That's all we're hoping for at the moment."