Leading Jewish organisations have called on Jeremy Corbyn to end the “impasse” over tackling antisemitism in the Labour party, calling the abuse received by Jews on social media during the row this summer “unfathomably vast”.

The Board of Deputies of British Jews, the Jewish Leadership Council (JLC) and the Community Security Trust (CST) wrote separate letters to Labour’s general secretary, Jennie Formby, accusing the leadership of failing to deal adequately with concerns.

The board’s chief executive, Gillian Merron, wrote: “While Labour could have used the summer to focus on any number of other serious challenges facing this country, the leadership has chosen to make its priority a fight with British Jews about antisemitism.

“The amount of abuse that Jews have received on social media from those purporting to support Jeremy Corbyn is unfathomably vast. While Mr Corbyn has said that such people do not speak in his name, his un-atoned-for actions undoubtedly give the wrong impression about what is permissible.”

Board of Deputies Chief Executive Gillian Merron has written to @UKLabour General Secretary Jennie Formby, informing her of the essential steps the party needs to take in order to deal with the antisemitism crisis pic.twitter.com/YjlWr3gdH1 — Board of Deputies of British Jews (@BoardofDeputies) August 29, 2018

Merron said the community had “no interest in an ongoing dispute with any major political party about the nature of racism against us”, but said the group had to defend the interests of Jewish people. “We hope that over the coming weeks and months, Labour will do the necessary to turn the tide and end this impasse,” she said.

The Guardian reported a fortnight ago that the party is preparing the ground to amend its antisemitism code of conduct to align with the International Holocaust Remembrance Alliance definition, following a consultation with Jewish groups, provided the party can include additional protections for criticism of Israel.

A crunch meeting of Labour’s national executive committee is to discuss the code on Tuesday after a bitter few weeks following the party’s decision to implement a code of conduct that failed to adopt four of the 11 examples of antisemitism given by the IHRA.

Labour proposed a further consultation to take place over the summer following an outcry from Jewish groups.

The party is now prepared to accept all the examples, including one the leadership considers most controversial – “claiming that the existence of a state of Israel is a racist endeavour” – but with caveats to allow members to describe actions of the Israeli government as racist and to criticise the circumstances of Israel’s creation.

Writing to Formby on Monday, the JLC and CST leadership said they did not feel able to respond to the latest consultation, pointing to the submissions the groups made to Shami Chakrabarti’s inquiry.

Today @JLC_uk and @CST_UK have responded to the Labour Party regarding the Code of Conduct on Antisemitism. Our view is that our submission to the Chakrabarti Inquiry in 2016 remains as valid then as it is now. pic.twitter.com/EeOsuPyu8H — The JLC (@JLC_uk) August 29, 2018

“Regrettably, we regard the current atmosphere as being too febrile and too lacking in trust for us to consult in any way that risks being construed as somehow lacking in transparency,” they wrote on Wednesday.

In her letter to Formby, Merron said the adoption of the IHRA definition alone would not be enough to rebuild trust, saying the party must take urgent steps to resolve outstanding disciplinary cases.

She also said there was significant concern over reports about Corbyn in the past few weeks, including comments at a conference in 2013 about British Zionists. Merron said the remarks implied that the Zionist group at the event were not British. “This is a classic racist trope,” she wrote.

The comments, which came to light last week, were made as Corbyn was defending a joke made by the Palestinian ambassador. The Labour leader said a group of Zionists had “no sense of irony” despite “having lived in this country for a very long time”.

Corbyn said he had used the word Zionist “in the accurate political sense and not as a euphemism for Jewish people”.

On Tuesday the former chief rabbi Lord Sacks called the comments the most offensive statement by a senior UK politician since Enoch Powell’s “rivers of blood” speech, a comparison Labour called “absurd and offensive.”

In their joint letter, the JLC and CST said Corbyn and the party must accept that most British Jews had close ties with traditional Zionism – support for a Jewish homeland – and Israel.

“The current obsessive hatred of Israel and Zionism means that no Jew can be an equal member of the Labour party, because even if they pass an initial loyalty/morality test, they still always remain under suspicion, lest they display ‘Zionist’ or ‘pro-Israel’ tendencies,” the letter said. “This is the current experience of Jews in Labour circles. It is an antisemitic environment.”

On Wednesday, Theresa May declined to say whether she believed comments by Corbyn about Zionism were antisemitic, but said Labour should address the concerns of the former chief rabbi and Jewish MPs in the party.

Speaking onboard her RAF Voyager plane en route to South Africa, the prime minister said Labour should take concerns of MPs such as Luciana Berger seriously.

Speaking later to broadcasters in Abuja, May said Sacks had “raised significant concerns, but it’s not just him – members of the Labour party have raised concerns as well. I think the leader of the Labour party needs to respond to those concerns”.

A Labour spokesperson said the party and Corybn “have made clear their total opposition to antisemitism and repeatedly stressed that we must ensure our party is a welcoming home for Jews, so that Jewish members can play a full part in our campaigning to rebuild Britain for the many not the few”.