Glenn’s remarks, criticised by a leading member of Britain’s Jewish community, came as he tried to justify the FA’s decision to charge Manchester City manager Pep Guardiola for wearing a yellow ribbon in support of jailed Catalan independence leaders.
Guardiola has until 1800 GMT on Monday to respond to the charge that he has breached FA regulations that ban the wearing of political symbols.
The FA has faced accusations of hypocrisy, given their prolonged campaign to persuade world football chiefs to allow players to display a poppy on their shirt to mark Armistice Day.
But Glenn, speaking following a meeting of the IFAB (International Football Association Board), football’s global law-making body, in Zurich, rejected the comparison.
“We have rewritten Law 4 of the game so that things like a poppy are OK but things that are going to be highly divisive are not,” Glenn was quoted as saying in British newspapers.
“That could be strong religious symbols, it could be the Star of David, it could the hammer and sickle, it could be a swastika, anything like (former Zimbabwe President) Robert Mugabe on your shirt — these are the things we don’t want.
“To be honest, and to be very clear, Pep Guardiola’s yellow ribbon is a political symbol, it’s a symbol of Catalan independence and I can tell you there are many more Spaniards, non-Catalans, who are pissed off by it.
“All we are doing is even-handedly applying the laws of the game.”
Simon Johnson, chief executive of the Jewish Leadership Council and himself a former director of corporate affairs at the FA, strongly criticised Glenn’s comment, saying the examples given were ill-judged and in poor taste.
“The Star of David (which features on Israel’s flag) is a Jewish religious symbol of immense importance to Jews worldwide. To put it in the same bracket as the swastika and Robert Mugabe is offensive and inappropriate,” he added.
Glenn issued an apology on Monday.
“I would like to apologise for any offence caused by the examples I gave when referring to political and religious symbols in football, specifically in reference to the Star of David, which is a hugely important symbol to Jewish people all over the world,” he said.
“I will be speaking with the Jewish Leadership Council and to (anti-discrimination group) Kick It Out to personally apologise.”
Glenn’s comments came just days after it was announced that Prince William, the president of the FA, would be the first senior British royal to make an official visit to Israel, later this year.
They are also bound to raise fresh questions about English football’s senior leadership after both Glenn and FA chairman Greg Clarke were widely criticised last year for their handling of bullying and racism allegations.