The UK’s new ambassador to the US has described Keir Starmer as having been “on the ropes” over the Peter Mandelson scandal and said it is Israel rather than Britain that has a “special relationship” with the White House.

Christian Turner, who took office in February to replace Mandelson as the UK’s most senior diplomat in Washington, made the remarks privately to a group of students visiting the US in the same month he was appointed.

His remarks are embarrassing for Downing Street because they emerged the same week that the king is carrying out his state visit under the president, Donald Trump, who has previously branded Mandelson a “really bad pick”. Mandelson was sacked by the prime minister last year for misleading him over the depth of his friendship with the late child sex offender financier Jeffrey Epstein.

Turner told the students it was “extraordinary” that the scandal “hasn’t touched anybody” in the US, while it had “brought down” Mandelson and “potentially the prime minister”, the Financial Times reported.

He said Starmer had at one point been “pretty clearly on the ropes” and his future had looked “quite touch and go” over the fallout from the scandal, adding that Starmer was a “stubborn guy” who would be unlikely to quit of his own accord. “The moment I would look to is the May elections,” Turner said.

“If Labour does very badly …  I suspect the party will be able to go over that threshold and remove him – seems to me to be the conventional thinking.”

He added: “If they do OK, he might carry on going … That’s just for me as a citizen speculating because I have to serve whomever is there.”

On the special relationship, Turner said it was “quite nostalgic, it’s quite backwards-looking, and it has a lot of baggage about it”, adding: “I think there is probably one country that has a special relationship with the United States – and that is probably Israel.”

Following publication of the remarks on Tuesday, a Foreign Office spokesperson said: “These were private, informal comments made to a group of UK sixth-form students visiting the US in early February. They are certainly not any reflection of the UK government’s position.”

A Whitehall source said the discussion was informal and focused on questions from students about diplomacy and the political issues of the day, and had clearly been never intended as an on-the-record statement of government policy. No 10 had no immediate comment on Turner’s remarks, which were unusually candid for a diplomat.

Turner was appointed as a supposedly safe pair of hands as a career civil servant and diplomat after the disastrous political appointment of Mandelson. He was chosen over Starmer’s business adviser, Varun Chandra, who instead took on an expanded role in Downing Street, and Nigel Casey, the ambassador to Russia.

Chandra was initially considered favourite for the high-profile post, but Olly Robbins, formerly the most senior civil servant at the Foreign Office, is understood to have lobbied against another political appointment. Robbins was dismissed by the prime minister earlier this month over his failure to tell him that Mandelson had failed security vetting – although Robbins maintains it was standard procedure to put in place mitigations and it was not as simple as a pass or fail test.

At the time of Turner’s appointment, the prime minister said: “I’m delighted that Christian Turner has been appointed to be British ambassador to the United States of America. The United Kingdom and United States have a very special relationship, and Christian’s extensive experience as an outstanding diplomat will support this uniquely close bond and ensure it continues to flourish.

“I warmly congratulate him as he starts his work to further build our strong economic and security ties and deliver for the British people.”

Turner was previously ambassador to the UN, and had been political director at the Foreign Office. He brokered a close relationship with the new Labour administration before taking up his UN role in New York.