Duncan, British Foreign Secretary Boris Johnson's second-in-command, and a long-time critic of Israel, was "doing a lot of problems," he added.
Duncan once described Israeli settlements in occupied Palestinian territory as a "stain on the face of the globe", calling them an "act of theft".
"Occupation, annexation, illegality, negligence, complicity - this is a wicked cocktail which brings shame on Israel," he told the BBC in 2014.
Masot also called Johnson an "idiot" and mocked "crazy" opposition Labour leader Jeremy Corbyn and his "weirdo" supporters.
The embassy issued an immediate apology, saying 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," it added.
Ambassador Mark Regev apologised personally to Duncan on Friday, according to the embassy. Britain's Foreign Office confirmed it had received an apology, and that it now considered "the matter closed".
The incident comes amid an Israeli diplomatic offensive against countries which backed a UN resolution last month demanding a halt to settlement building on Palestinian land.
All 15 members of the UN Security Council, including permanent member Britain, voted in favour of the resolution except for the United States which abstained, sparking Israeli fury.
It was not immediately clear if Masot's comments were linked to the resolution. Conservative chairman of the Foreign Affairs Select Committee Crispin Blunt told the Mail on Sunday the comments were "outrageous".
The footage was recorded at a restaurant opposite the Israeli embassy in London's upmarket Kensington district, during a meeting attended by Maria Strizzolo, senior aide to Conservative Education Minister Robert Halfon.
Copyright Agence France-Presse, 2017