A banner held up outside Parkhead on Wednesday morning, following a 2-2 draw with Hibernian at the weekend, read, "Save the ten - time to go Neil".
The Green Brigade, the ultras section of the Celtic support, also released a statement which called for chief executive Peter Lawwell and the board to act quickly to prevent the chance of a record-breaking 10 successive Premiership titles being missed.
Celtic have just two wins out of their last eight games in all competitions and sit 11 points behind Premiership leaders Rangers, albeit the champions have two games in hand.
Ahead of the Europa League clash with Sparta Prague on Thursday, veteran midfielder Brown said: "I think in today's age it is all social media, a picture for Instagram or Twitter or whatever.
"We have 100 per cent faith in the manager.
"The support that we have had through the nine in a row has been incredible.
"I think now is the time we all come together. The fans have been fantastic for us for the last nine season and this is the time we all stick together.
"We're pushing for something special. We need to make sure that we are all together.
"Everyone in that changing room is working extremely hard to get better.
"We know that performances haven't been up to scratch and now we have to fix it and we need to make sure we fix extremely quickly because the manager and backroom staff have been working extremely hard.”