WASHINGTON — Saudi Arabia paid millions of dollars to Washington lobbyists to keep it out of court. They have been unsuccessful. And now it is up to the kingdom’s lawyers to limit the damage.

With families of Sept. 11 victims now able to pursue legal claims against the Saudis, the fight over responsibility for the terrorist attacks 15 years ago is likely to shift to a courtroom in Lower Manhattan, not far from where the World Trade Center once stood.

The legal battle could last for years, and would be waged using thousands of pages of documents, deposition transcripts and official government investigations. It could end in millions — or billions — of dollars’ worth of Saudi assets being seized in a court settlement, or a judgment that largely vindicates the Saudi government, which for years has insisted it had no role in the deadly plot.

Lawyers for both sides were shaping a legal strategy on Thursday, the day after Congress overrode a veto of a law allowing the 9/11 suits to go forward. For more than a decade, they have been blunted by a sovereign immunity law protecting foreign governments from American lawsuits.