Ally McCoist's side are guaranteed to be drawn against an SPL side in the last eight, with supporters groups, including the Rangers Supporters Association and Assembly, calling for the Light Blue legions to snub the fixture if it is away from Ibrox.
Many fans are still furious with Rangers being refused entry into the SPL, and supporters groups are ready to make themselves heard.
John Macmillan of the Rangers Supporters Association told SportTimes: "There is a feeling out there that this is the right thing.
"I have had emails and phonecalls from members saying they are 100% behind this. We decided on this at our AGM, it is not something we have just thought about in the last couple of weeks.
"It was our members who voted to have a boycott. It wasn't a case of me or a committee saying 'we are not ordering tickets'.
"We would hope that the club would support this. If the club doesn't support it and decides to take tickets then that makes the boycott a little more difficult for our fans.
"They may say 'well, Rangers have taken tickets so I will'. They will make their own decision. We would hope Rangers would support this.
"For a one off, I think it would be worth it and I also hope Rangers and Charles Green support us in this."
A 30,000-strong crowd cheered Rangers to a 2-0 win over Motherwell in the last round as McCoist's side took their first SPL scalp since being placed in the Irn-Bru Third Division.
And Macmillan added: "In my view, this is a boycott of whatever team we draw if it is away from home, no matter who it is.
"That may hurt a lot of people, who, week in and week out, follow the team everywhere.
"I know it is a difficult decision but I would urge people to make the right one. It is an attempt to show unity, and I am sure we have that, and our feelings against these clubs.
"This is the first opportunity we have had to do it. There were supporters who felt we should have boycotted the Motherwell League Cup tie at Ibrox but that is different because it is a home game.
"I would find it very difficult to persuade people to boycott a home game."