WASHINGTON — A bill that would let the families of those killed in the Sept. 11 attacks sue Saudi Arabia for any role in the terrorist plot passed the Senate unanimously on Tuesday, bringing Congress closer to a showdown with the White House, which has threatened to veto the legislation.

The Senate’s passage of the bill, which will now be taken up in the House, is another sign of escalating tensions in a relationship between the United States and Saudi Arabia that once received little scrutiny from lawmakers.

Administration officials have lobbied against the bill, a view that the White House spokesman Josh Earnest reiterated after the vote. And the Saudi government has warned that if the legislation passes, it might begin selling off up to $750 billion in Treasury securities and other assets in the United States before they face a danger of being frozen by American courts. Adel al-Jubeir, the Saudi foreign minister, delivered the warning to lawmakers and the administration while in Washington in March.

Many economists are skeptical that the Saudis would deliver on such a warning, saying that a sell-off would be hard to execute and would do more harm to the kingdom’s economy than to America’s.