The Magpies manager is under intense pressure following a dreadful start to the new Premier League season.
Newcastle are winless in their opening four games, with just a single draw and three defeats, after an uninspiring summer transfer window.
Bruce has been unpopular at St James' Park ever since his 2019 appointment, even amid more positive results last season.
Tensions have now reached boiling point, with fans returning to vent their fury and regular clashes with local journalists.
And speaking on talkSPORT earlier this week, former Crystal Palace chief Jordan, a former employer of Bruce, urged him to leave.
"A lot of Newcastle supporters don't like Steve Bruce - they don't rate his record," said Jordan on Tuesday.
"I happen to rate Steve Bruce, I think he's a decent football manager.
"I think he should leave Newcastle, I don't think he can ever turn this around, I don't think he's ever going to right this.
"I think he should quit? Oh, absolutely, and I've told him unequivocally."
He continued: "The time to quit was when he did his job last season."
But Bruce, a boyhood Toon fan, has no intention of taking that advice.
When asked about Jordan's comments, the Magpies boss delivered a defiant response.
Bruce said: "I understand where Simon is coming from, but I never really thought about it then.
"I never thought just because it's tough here, I'm going to walk away."
He added: "I believe with the experience that I've got that I can try and help the club. A lot of people will find that laughable - I understand.
"But that's what I genuinely believe. I hope that my experience can see us through."
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