The United States had a very interesting role in the formation of Israel, most importantly because of a sort of disconnect between different parts of the government.
In 1919, the King-Crane commission was formed by the United States. It was meant to investigate what should be done with all former Ottoman territories, in accordance with the wishes of the locals. When it came to establishing a Jewish state, the Commission recommended against it, stating it would be difficult if not impossible, and would result in human rights violations. The Commission also recommended that the United States assume occupation of the area, arguing that a colonial government would be improper and that independence would be likewise improper. Some historians note that the opinion about the formation of a state for Jews may have been influenced by Crane's anti-Semitism, and Crane would later express admiration for Adolf Hitler and his treatment of Jews.
Still, the opinion persisted that a Jewish state would not be ideal in many diplomatic posts in the United States. Over time, however, Zionist groups had managed to gain significant influence in the United States by spreading their message and establishing branches, much in the way that other grassroots initiatives did at the time. The grassroots support and support of the wealthy was so strong that Golda Meir (later Israel's Prime Minister, in the 1970s) raised $100 million between January and June in a tour of the United States, funds which were crucial to arms payments to help Jewish forces win the civil war that went on from November 1947 until the May 1948 invasion by the Arab states.
Even so, the fight was mostly had in the mid-1940s until the moment Israel was formed, and perhaps even a little after. In 1944, FDR convinced Congress not to push forward with a joint resolution calling on the British to rescind the 1939 White Paper (which had limited Jewish immigration and said that within 10 years a single state, likely Arab, would be formed) and support a Jewish state. Even so, and despite promising King Ibn Saud (Saudi Arabia) that Jews and Arabs would both be heard (and Arab populations didn't want a Jewish state anywhere near them), Roosevelt also promised the supporters of a Jewish state that "full justice will be done to those who seek a Jewish national home...".
We'll not know what FDR would've definitely done, though we do know he was still telling Ibn Saud that he would not support any action that proved "hostile to the Arab people", and that Zionism (support for a Jewish state) was proving increasingly popular in the United States and among US officials. But when FDR passed away, Truman (who was moderately more in support of Zionism) suggested settling Jewish displaced persons (DPs) from WWII in the region. The Arab League Secretary General, in response, said this could start a "new war" between Christianity and Islam, so as you can see it wasn't well-received.
With the voting out of Churchill, Prime Minister Atlee was voted in for Britain. His foreign secretary proposed continuing the restrictions on Jewish immigration to 1,500 a month, despite the expiration of the White Paper of 1939, which Truman opposed. Truman forwarded to Atlee a report saying that 100,000 Jews should be granted entry permits to get them out of displaced person camps in Europe. Truman was persuaded not to say his support publicly, but by October 1945 his support came out. The British saw themselves between a rock and a hard place: they didn't want to harm the "special alliance" between Britain and the US, but they also didn't want to enrage the Arab world by allowing Jews to enter the region of Palestine.
So the British chose to stall a bit longer, establishing yet another commission with the Americans. Following up the Peel Commission (1937) and the Woodhead Commission (1938), among others, this one was tasked with deciding what to do with Jewish displaced persons. Six British men and six American men studied the situation of the displaced persons, toured the Middle East, read reports, and heard testimony. They submitted a thousand-page volume. Their tours were quite choreographed by both sides. The Zionist leadership made sure they only met with Zionist Jews in the displaced person camps, while people like the Saudi king spoke of how "...the Jews are our enemies everywhere. Wherever they are found, they intrigue and work against us...", before presenting each commission member with a golden dagger, an Arab robe and headdress, and he offered (perhaps jokingly) to find the commission's chair a wife.
The Commission eventually was swayed far more to the Zionist side. Quoting one of the books I'm referencing:
One American committee member, Frank Aydelotte, later wrote: ''I left Washington pretty strongly anti-Zionist...But when you see at first hand what these Jews have done in Palestine...the greatest creative effort in the modern world. The Arabs are not equal to anything like it and would destroy all that the Jews have done...This we must not let them do."
Frank Buxton, another American member, wrote: "How my Vermont father...would have been amazed at the greater deeds of the Palestinian Jews...I came away from those farms less cocky and more humble and not quite so certain that American pioneers left no successors."
The commission had a few recommendations. First, it recommended immediate permit-granting to 100,000 Jews to enter the region. Second, it said that partition into two states was unworkable, and a single state should be established, but that the single state was not ready and that the UN should be trustee for a while longer. Truman supported the 100,000 permits, while again Britain's prime minister said he opposed it until the Jewish leadership was disarmed (Jewish groups at the time were carrying out a guerrilla campaign against British forces).
A steady process led, finally, to Truman endorsing partition and establishment of a Jewish state in October 1946, and he supported "substantial" immigration immediately. Britain, as a result, abandoned the Mandate, since it no longer had even an Anglo-American understanding on the future of it.
So the United Nations, with the support of the United States and (surprisingly) the support of the Soviet Union, established a committee on Palestine called the UN Special Committee on Palestine (UNSCOP). The committee, long-story short, came up with an idea for essentially a federal state (the minority report, supported by less than half of the committee members), and for partition (the majority report, supported by more than half of the committee members).
The State Department didn't support the majority report, or partition. It was actually fairly opposed, and Truman was at odds with them: he supported partition, and reaffirmed that support. The Soviets were behind it, he had supported it before, and he would support it again. So began intense lobbying on the issue, and attempts to get enough votes to pass it. Truman interceded on the behalf of the Zionist groups two times most importantly, in my view: on the Negev, and on the whole vote. When there was a proposal to remove the Negev desert from the Jewish state's allocated borders, Truman opposed it and it was shut down. But as the vote drew close, Truman's order to his delegation at the UN not to intervene or try to strongarm any other delegations into voting for partition stood in place. With last-minute pleas mounting by Zionists, and their concern that the Arab-Muslim bloc (which included Iran, Lebanon, Pakistan, Saudi Arabia, Syria, Yemen, Afghanistan, Egypt, and Turkey) made up more than 20% of those that did not abstain. In fact, the only non-Muslim and non-Arab countries who voted against were India, Cuba, and Greece, thanks to the strong-arming of the United States. The United States was more convincing than threats made by the Arab states, as well as some reported attempts at bribes by Arab states. There were accusations of bribes either way, of threats, and more. It was quite a mess.
In the end, the plan passed. Had three "yes" votes gone "no", it would not have passed. But this plan did not establish Israel, contrary to popular belief. It merely recommended partition, and asked the United Nations Security Council to implement it. It never did. Instead, a civil war broke out with the Jewish groups celebrating and the Arab groups fuming. This civil war would usher in the next play by the United States, which was more of a "hands-off" approach.
The civil war prompted an arms embargo by the United States, and then by the broader United Nations. This embargo would have harmed the Jewish population far more (because Arab states still had arms deals that would be delivered on), had not the embargo failed to stop arms-running by Jewish groups. In addition, Czechoslovakia signed a crucial arms deal with Israel (once founded on May 15, 1948) that essentially kept it fighting and helped even the field in terms of weaponry.
The United States did one more crucial thing: Truman, against the advice of his State Department, recognized the new state of Israel de facto within 11 minutes of its declaration of independence on May 15, 1948. That provided it with serious legitimacy. This was a weaker support than de jure recognition, which waited until the end of the 1948 war, but it was still crucial for Israel to get some legitimacy. Not to be outdone, the Soviets recognized Israel de jure three days after its declaration of independence.
Do you have more questions about the US's role in the formation process? I think I covered the major points, but I may be rushing a bit to finish explaining.