Rangers Supporters' Trust has warned there could be further demonstrations if results and displays do not improve.
Alex McLeish avoided the axe earlier in the season
Around 300 fans demonstrated outside Ibrox after Saturday's 3-0 home defeat by Hibernian in the Scottish Cup.
And Trust spokesman Stephen Smith said: "Regrettably, the protests will continue and will come after the next bad performance or result.
"There is just such a malaise there now. We don't organise the protests, but there is a lot of discontent."
Rangers have reached the last 16 of the Champions League - having been the first-ever Scottish side to qualify from the group stages.
But reigning champions Rangers lie a distant 15 points behind Scottish Premier League leaders Celtic and are now also out of the Scottish Cup and CIS Insurance Cup.
And Ibrox chairman David Murray and manager Alex McLeish were both targeted by the protest after the fourth-round defeat.
"You can understand why they are happening because supporters need clarity over how the club is going to turn itself around," Smith told the Daily Express.
"If Celtic batter Rangers on Sunday, or bad losses are suffered to Villarreal or Hibs, the same will happen.
"It's time for the chairman to say what the situation is for the rest of the season."
McLeish survived much speculation about his position earlier in the season after Murray appeared to be searching for a replacement.
The chairman eventually stood by his man and said he wanted to add three players to McLeish's squad during the January transfer window - a defender, left-sided midfielder and a striker.
But striker Kris Boyd, from Kilmarnock, was Rangers' only capture.
"We were led to believe we would get three players and they didn't arrive," added Smith.
"What has happened to the big investment that was promised as well?"