The British Council withdrew from an iftar organised by the Ramadan Tent Project a week after Labour Party leader Sir Keir Starmer pulled out of a similar event over concerns about the RTP's links to controversial advocacy group Cage.
James Hampson, a director at the British Council’s UK office, was to attend the online gathering on Thursday.
The British Council has since said Mr Hampson will not take part.
“The British Council operates in over 100 countries and is respectful of all religious faiths," it said.
"The British Council is an apolitical organisation and for that reason we are unable to respond further."
RTP said Mr Hampson was no longer able to attend because of last-minute commitments. It says it is “on a mission to bring communities together to better understand one another”.
The Jewish Chronicle reported that Mr Starmer, leader of the opposition in the British Parliament, pulled out of last week's event after comments made by Omar Salha, chief executive of the RTP, were brought to the attention of the Labour Party.
“Time for those who stand for the values of justice and liberty to support UK Cage and not be intimidated by the powers that be,” Mr Salha tweeted in 2017, in reference to the advocacy group.
Cage drew controversy in 2015 when its research director described Mohammed Emwazi, a British ISIS fighter who beheaded western hostages in Syria and made propaganda of the murders, as a “beautiful young man”.
Emwazi was in contact with Cage before he went to Syria and after he said he had been harassed by the intelligence services.
Mr Salha has shown support for a boycott of Israeli goods.
RTP has teamed up with Islamic Relief, which is the charity partner of the iftar's.
Despite Mr Starmer withdrawing last week, Andy Burnham, the mayor of Manchester and a member of the Labour Party, was a guest at a similar event on Wednesday. Also present was Zia Salik of Islamic Relief.
Islamic Relief attracted controversy last year when it was discovered that a former director had made anti-Semitic remarks on social media. The charity has since pledged to clear up its act.
“This is clearly a troubling development, regardless of what political party is involved,” said Ghanem Nusiebeh, chairman of Muslims Against Anti-Semitism.
He said Cage and Islamic Relief have been linked to anti-Semitism and extremism.
“This coming after the anti-Semitism scandal in Labour and involving Muslims not only sends the wrong message on whether Labour is serious about tackling anti-Semitism, but creates an atmosphere of normalising anti-Semitism within the British Muslim community."
In response to the Jewish Chronicle article, the RTP said "any attempt to undermine the excellent relationship Ramadan Tent Project has with the British Jewish community is divisive and deeply concerning".
"Ramadan Tent Project has always sought to welcome people from all backgrounds and our mutually respectful and positive relationship with the British Jewish community is an important constituency which falls within that."
The concerns were first raised by Tal Ofer, a member of the Board of Deputies, a representative body of the UK’s Jewish community.
The RTP’s Twitter account praised Muslim Engagement and Development, the organisation more commonly known as Mend, which faced criticism criticised for campaigning against the government’s counter-extremism policies.
The Henry Jackson Society think tank said in a 2017 report that Mend regularly hosted illiberal and intolerant speakers at public events.
“Instilling the spirit of Islam all year round, our events are open to all faiths and none, aiming to soften hearts and minds and bring about good relations between people for a more harmonious society,” the RTP's website says.
Mr Burnham’s office has been approached for comment.
