Absolutely! This is a really great question. Not a lot of people are aware that while there is a conspiracy theory about Jewish or Israeli influence historically in the United States, Israel (and many other countries) have often claimed American interference in their domestic affairs.
Perhaps the best place to start would be the most recent example of this that we can talk about in the 20-year rule. The 1999 election in Israel was contentious, but unfortunately falls just outside of our sphere, so I won't go into that. I'm a strict adherent to that rule. I try not to bend it.
Instead, I'll look to 1996, where Prime Minister Benjamin Netanyahu won the Prime Minister spot in one of the most contentious elections in Israeli history. This election, which took place in the unusual (and soon-ditched) system of voting separately for the Knesset (Parliament) and for the Prime Minister, pitted Netanyahu against the acting Prime Minister, Shimon Peres. This system was different, because typically the Prime Minister is selected from within Parliament, usually by the party that gets the most seats (or, sometimes, from the party that can form a coalition with the seats/allies it has, if it comes in second, and the top party cannot form a coalition government). 1996 was the first time that the Prime Minister slot was allocated by a separate ballot, which made the election far more about personalities of leaders than prior elections.
This election also followed the assassination in 1995 of Yitzhak Rabin by a far-right Israeli named Yigal Amir, who opposed the peace process with the Palestinians. The assassination of Rabin shocked the nation, and was based on a unique (and, most Jewish scholars agree, absurd) set of reasoning pushed by a fringe subset of rabbis that permitted violence against a "rodef" (pursuer). Amir, and some fringe theologians, argued that Rabin was a "rodef", because his goals would bring harm to other Jews, and therefore this was justification for his murder. Suffice to say, this was not the opinion of the vast, vast majority of Jews or Jewish scholars, and Rabin's assassination added an even more challenging wrinkle to the election as far as its emotional value and impact, particularly since, Peres was Rabin's Minister of Foreign Affairs and one of his closest allies. Those familiar with the Oslo Process may recall this photo of Rabin and Arafat shaking hands, related to the signing of the Oslo Accords. This photo, from the same event, shows Peres himself at the table, as Rabin, Arafat, and President Clinton stand back as well. Peres was integral to Oslo, and the 1996 election was seen as integral to the continuation of the peace process as well.
All of this is to lay out the setup for this election. But there's yet another wrinkle: Netanyahu himself was accused of aiding, or at least ignoring, incitement that may have encouraged Yigal Amir. It's unclear how much of a direct impact that could have had on Amir, and chances are it wasn't much, but the general air in which the Israeli right-wing parties conducted themselves, and the speeches of some of their officials, were quite incendiary and could have been worsened by their speeches. Netanyahu himself claims he was not inciting any violence, particularly not against Rabin. He points to speeches he gave specifically stating that violence was not the solution to political differences, saying in one rally to a crowd chanting that Rabin was a "boged" (traitor) in April (the assassination was in November) that:
He [Rabin] is not a traitor. He’s not a traitor. He’s mistaken. He’s greatly mistaken. And he will end up stepping aside. But he’s not a traitor. He is not a traitor. No, no. No, he is not. We are dealing with political rivals, not enemies. We are one nation.
In October 1995, a news report stated that:
The crowd chanted ‘Rabin Traitor’ and Mr. Netanyahu silenced them, saying such calls do not belong here. ‘We will not replace the government through fire and blood but through the ballot box, and through it alone.
Even so, Netanyahu's general rhetoric and the inflammatory scene in general, as well as the sense that Netanyahu was fanning the flames behind the scenes, led to a lot of bad blood going into the election.
Enter America.
The President, Bill Clinton, had a clear preference. He makes no secret of it today, saying that he tried to help Peres win in 1996, without being too "overtly involve[d]". Clinton liked Peres, and thought Peres would be more supportive of furthering the peace process. That's not to say that Clinton didn't work with Netanyahu whenever possible after Netanyahu won. But that is to say that Clinton's preference manifested itself in ways that attempted to tip the scales. Netanyahu was aware of it too, Clinton says. That's just how things went. Relations were always tense after that between the two men, despite some minor achievements. So what did Clinton do? It's unclear how personally involved he was in the actions, but some hackles were raised over the positioning of Peres by Clinton. Clinton organized an international peace conference of sorts to boost Peres's standing, and praised him profusely during the lead-up to elections. He pledged new US aid to Israel, and they coordinated public messaging.
Read my reply to this comment to continue.