In a secretly-recorded conversation, The Israeli embassy's senior political officer Shai Masot said Minister of State Sir Alan Duncan was causing "a lot of problems".
The Israeli ambassador to the UK has since apologised for the incident and said the remarks don't reflect the views of the Government of Israel.
The conversation was recorded at a restaurant opposite the Israeli Embassy in Kensington during October last year, as part of an investigation by Al Jazeera.
Those at the meal were Mr Masot, the undercover reporter and Maria Strizzolo, an aide to education minister Robert Halfon, the former political director of Conservative Friends of Israel.
The reporter posed as a pro-Israel graduate activist. Mr Masot asked Ms Strizzolo: "Can I give you some names of MPs that I would suggest you take down?"She replied that all MPs have "something they're trying to hide".
Mr Masot responded by saying: "I have some MPs... She knows which MPs I want to take down... the deputy foreign minister".
Sir Alan has previously criticised Israel and was seen as a bigger problem than his boss Boris Johnson who was dismissed by Mr Masot as "basically good" but also "an idiot".
A transcript of the conversation published by the Mail on Sunday had Mr Masot saying "(Boris Johnson) just doesn't care. He is an idiot but has become minister of foreign affairs without any responsibilities. If something real happened, it won't be his fault... it will be Alan Duncan."
The Israeli Embassy said in a statement that it "rejects the remarks concerning Minister Duncan, which are completely unacceptable; the comments were made by a junior embassy employee who is not an Israeli diplomat, and who will be ending his term of employment with the embassy shortly."
