Liz Truss, the Conservative leadership frontrunner, has been criticised after accusing the civil service of having a "woke" culture that "strays into antisemitism".
The foreign secretary also referred to "setting up your own business" as being a "Jewish value".
Speaking on Sky News, Steven Littlewood, the assistant general secretary of the civil service union the FDA, said: "Ms Truss has already accused [civil servants] of being lazy, threatened to cut their pay.
"But to be accused of institutional antisemitism - this is essentially what she's accusing them of - I mean, they're absolutely astonished.
"I've had members get in touch with me today saying they're upset, that they're appalled.
"I've had Jewish members contact me saying that they feel really uncomfortable about being drawn into the middle of a political controversy and used to attack the civil service."
FDA general secretary Dave Penman said Ms Truss provided "no evidence for her accusation", which he said "goes further than the usual dog-whistle politics" of the ongoing Conservative leadership election.
"The Conservatives have been in government for more than 12 years now and, for most of that time, Liz Truss has been a minister," he said.
"So accusations of 'civil service wokeism' are a little ironic, given it's essentially a criticism of their own leadership."
In a joint letter with Prospect general secretary Mike Clancy, the pair called on Ms Truss to provide examples of antisemitism to the cabinet secretary.
They said: "Unfounded allegations designed simply to generate headlines is an approach they are all too familiar with, and one they will have hoped would end with a change of prime minister.
"If you are able to make such serious allegations, you have a duty to ensure they are backed up with evidence. We would therefore suggest that such evidence is made available as a matter of urgency to the cabinet secretary for investigation."
In a written statement issued after she spoke at a synagogue in Manchester, Ms Truss referred to "wokeism" in the civil service, adding: "Every organisation has its culture, but it's not fixed, it can be changed.
"That's what ministerial leadership is about: it's about making sure that the policies we represent, the values we stand for, are reflected in what we do.
"I've been very clear with our officials about the positions we take on Israel, and that will continue if I become prime minister."
It is understood the pledge relates to a suggestion Ms Truss needed to "overrule" Foreign Office officials who had voiced doubts about her show of support for the Israeli prime minister at the UN Human Rights Council.
In an interview, she told the Jewish Chronicle she had stood alongside her "good friend" Yair Lapid, despite some in the department claiming this could lead to Britain becoming "isolated".
The foreign secretary has also pledged to review whether schools are doing enough to educate pupils and teachers about antisemitism, according to her team, and said university campuses must be "ridded" of the issue.
"So many Jewish values are Conservative values and British values too, for example, seeing the importance of family and always taking steps to protect the family unit; and the value of hard work and self-starting and setting up your own business," she said.
"The British Jewish community is incredibly proud of this country and so are Conservatives."
Labour MPs, including some who themselves are Jewish, also criticised the remarks.
Margaret Hodge, who campaigned against antisemitism in the Labour Party, said: "Using antisemitism to peddle the right's 'anti-woke agenda' is below the belt. The oldest form of racism is not a tool to use in the divisive culture war nonsense."
A source for the foreign secretary's campaign claimed that Labour has been a "talking shop for antisemitism and anti-Zionism" under the leadership of Sir Keir Starmer and Jeremy Corbyn.
"This has increased anxieties within the Jewish community," they said.