Gary Rowett hit out at Stoke disgruntled fans after his side's 0-0 Betfred Championship draw at Bolton.
Wanderers' fourth point of the Christmas period lifted Phil Parkinson's side out of the bottom three.
But City supporters booed their team and chanted Bojan Krkic's name during and after a game that leaves the Potteries outfit seven points off the top six.
"They can chant what they want at me," said the Stoke boss. "That's fair enough.
"I wouldn't criticise them for giving me some stick either about the football or the amount of wins we have got.
"But to boo your own team after 20 minutes is not helpful. If I say it and I get criticism, I am not really bothered.
"We had control of the game for the first 20 minutes and passed the ball well. Our first backward pass and they boo.
"That's not helping their own team and the one they support. I understand the frustration and I would rather aim their frustration at me.
"If we don't win enough games we all know what the reality will be. The only way to help your own team on the pitch is to get behind them."
On the ongoing support for Spaniard Bojan, who last played on November 24, Rowett added: "He has been a fantastic player for the club.
"But that was two or three years ago and we have got to move forward.
"I am surprised they didn't chant Rory's (Delap) name because he was a fantastic player for the club as well."
On his side's failure to breakdown Bolton, Rowett added: "We weren't good enough in the final third. That's been our issue.
"It's another game we had reasonable amount of control of but it's no good if you don't turn it into chances.
"Until we score more goals and until we create more opportunities we are not going to win enough games."
Bolton boss Parkinson was delighted his team followed up a 2-1 Boxing Day win over Rotherham.
"Our aim is to still be out of the bottom three by the time the FA Cup third round (at home to Walsall) comes round," he said. "We have one more game at Hull to achieve that.
"I am so pleased with the manner of the display. It was a really disciplined performance.
"We could have won but there are still a lot of plus points. We have achieved a clean sheet against a side that spent £40 million in the summer.
"I felt the standout performances all had white shirts on."
England keeper Jack Butland denied Wanderers a deserved win with stunning saves from Christian Doidge and Mark Beevers.
"Stoke being able to keep hold of him last summer was huge for Gary," added Parkinson. "He kept them in the game.
"It was important we backed up the Rotherham result and I am so pleased with the concentration but also the energy levels because it was mentally and physically demanding against Rotherham."
