WASHINGTON — President Obama is delaying a planned veto of a bill that would allow the families of victims of the Sept. 11 attacks to sue Saudi Arabia for any role in the plot, hoping to tap into an unusual well of buyer’s remorse among senators who passed the measure unanimously in the spring.

The measure sailed through the House last week after a surprise last-minute vote, raising the prospect of the first veto showdown between Mr. Obama and a bipartisan coalition in Congress. But an intense lobbying campaign by the White House and Saudi Arabia, among others, has cast doubt on what had appeared to be an inevitable override of the president’s long-expected veto.

Officials have refused to say when Mr. Obama would veto the bill, and he has until next Friday to do so. His advisers are considering whether he should wait until then, after Congress is expected to recess on Thursday for the November elections, which could give him weeks to persuade lawmakers to drop their support for the measure before they return and consider the veto override.

Already, cracks are showing, even among Republicans who generally would love to exercise the first veto override against Mr. Obama.