But the pressure on the former prime minister has only emerged in recent weeks, since he announced his return to front-line politics last month. Mr Epstein was arrested last week.
Scrutiny has intensified following a report in Haaretz, confirmed by Mr Barak, that Mr Epstein had invested a $1 million (£800,000) in a high-tech company founded by Mr Barak in 2015.
Another possible financial connection between the two is the $2.3 million (£1.84 million) that Mr Barak received for undisclosed research in 2004 from the Wexner Foundation, which provides professional development for Israeli civil servants and Jewish-American community leaders.
Its founder, Les Wexner, worked closely with Mr Epstein — but according to a statement put out by the foundation last week, the two parted ways 12 years ago.
On Tuesday, the Daily Mail published new photographs taken outside Mr Epstein’s home in which Mr Barak, his face partly hidden by a muffler, was seen entering, apparently on the same day that a group of young women were seen there as well.
Mr Barak has instructed law firm Mishcon de Reya to demand the newspaper apologise for “suggesting that he is somehow involved in ‘Jeffrey Epstein’s pedophile ring’,” remove the report and pay him unspecified damages.
The details have already been alluded to in Likud’s election campaign through a video asking: “What else did the sex offender give Ehud Barak?”
Mr Barak responded by saying that he knows both Mr Epstein and Benjamin Netanyahu well, and hopes the truth about the allegations against the two of them will come out soon.
In interviews, he claimed to have been first introduced to Mr Epstein 17 years ago by the late President Shimon Peres and that he had met him on at least ten occasions, but never at parties or in the company of young women.
Mr Barak refused to elaborate on the research he had done for the Wexner Foundation, saying it was confidential, but insisted that the payment was perfectly legal and was reported to the tax authorities in Israel. He added that he is looking for ways to divest Mr Epstein from partnership in his company.
Even before the latest reports, Mr Barak’s Democratic Israel party was failing to find buoyancy in the polls. Most projected between four and six seats and, so far, neither the leader of Labour nor Meretz seem very eager to create a joint list ahead of the September 17 election.
