Real head coach Zinedine Zidane has come under mounting pressure amid a run of form that has seen the club win just one game out of their last five.
The defending Spanish and European champions are in fourth place in La Liga, 16 points adrift of leaders Barcelona and just a point above fifth-placed Villarreal.
Gibson said it would be a mistake for the Madrid hierarchy to dispense with Zidane midway through the season and called for new signings to freshen up the squad.
"I don't think they should make the change," he told Sky Sports. "There are certainly changes needed within the squad.
"There are issues but clearly I wouldn't sack Zidane. I don't think it's the right time for Real to be sacking the manager. I think everybody at the club needs to pull together and they have to make some big decisions."
Gibson suggested that Real's first-team players had grown complacent due to there being no serious threat to their place from the bench.
"Perhaps not in this window, but there needs to be measured and considered decisions on players that leave the club and players that come into the club," he said.
"The chequebook has to come out again. They have to back the manager and the manager has to be prepared to bring in big-name players. If his squad is bigger than it is now and he's upsetting players again, as was the case last season hence why James Rodriguez, Pepe, Alvaro Morata and Danilo left, then that has to be the case.
"Real Madrid is arguably the biggest club in the world and you shouldn't have an established 11 that are under no pressure for their positions. When you look at Real's bench at the moment that's exactly what is happening."
Real travel to Leganes for the first leg of the Copa del Rey quarter-final on 18 January.