The 700 plaintiffs said on Tuesday that they were seeking billions of dollars in compensation from the Jordan-based bank.
Lead counsel Ron Motley said he wanted to see "a verdict big enough to change their policies".
The lawsuit accuses the bank of promoting terrorism by paying money to the families of resistance fighters run by Palestinian group Hamas.
The legal action was brought under the Alien Tort Claims Act of 1789, which grants non-US citizens access to the US courts to seek justice for violations of "the law of nations" such as genocide and crimes against humanity.
One of the largest financial institutions in the Middle East, the Arab Bank has more than 400 branches and offices in 25 countries, including a branch in New York City.
Motley plans to tell the court that the Bank "aided and abetted the recruitment of genocidal murderers and was allegedly funding anti-Israeli occupation activities".
One of the plaintiffs, Iris Almog Schwartz, an Israeli citizen whose family members died in the bombing of a Haifa restaurant in October last year, said the US courts were a powerful weapon.
"Today, we are striking back non-violently with the most powerful weapon we have - the US courts," he said.
Kevin Walsh, a lawyer with Winston & Strawn LLP, said on behalf of the Arab Bank:
"Based on the press release issued today, which is all we have seen so far, Arab Bank asserts unequivocally, and above all else, that it deplores and condemns terrorism in all its forms, and is saddened by the consequences of it.
"The accusations being brought against the bank, as we understand them, are entirely false. The bank has never and would never support terrorist organisations in any way. The bank in due course will defend itself vigorously."
Since the start of the al-Aqsa intifada, more than 980 Israelis have been killed - including more than 100 children - and more than 6700 have been injured.
But in the same period, more than 3490 Palestinians have been killed - including more than 650 children - and more than 28,280 have been injured.