"I think the will of the American people is consistent with letting it expire, so it will expire," Senate Majority Leader Bill Frist, a Tennessee Republican, told reporters.
The 10-year ban, signed by President Clinton in 1994, outlawed 19 types of military-style assault weapons. A clause directed that the ban expire unless Congress specifically reauthorized it.
Some Democrats and several police leaders said President Bush should try to persuade Congress to renew the ban. Mr. Bush has said he would sign such a bill if Congress passed it.
"If the president asked me, it'd still be no ... because we don't have the votes to pass an assault weapons ban and it will expire Monday and that's that," House Majority Leader Tom DeLay, a Texas Republican, told reporters later.
DeLay said the ban was "a feel-good piece of legislation" that does nothing to keep weapons out of the hands of criminals.
However, House Speaker J. Dennis Hastert, an Illinois Republican, said he would consider allowing the House to vote on legislation only if the Senate acted first.
Appearing at a news conference, chiefs of police from the District of Columbia, Los Angeles, Atlanta and Seattle predicted an increase in violent gun crimes if the bans does expire.
"Our streets, our homes, our citizens and our police officers will face great danger unless the federal ban on assault weapons is renewed," said Charles H. Ramsey, the police chief in the nation's capital.
In March, the Senate voted to add the ban to a bill that would have immunized gun manufacturers from liability suits stemming from violent gun crimes. But the Senate voted 90-8 against the final bill after the National Rifle Association urged its defeat.
NRA executive vice president and chief executive Wayne LaPierre said in an interview with The Associated Press that his group is so confident that Congress won't renew the ban that it is not spending any more money on ads this year opposing it.
He said supporters of the ban could not muster the support needed to bring it to a vote in the House because several Democrats attribute losing their majority in the House in 1994 over votes then in favor of the ban.
Opponents of the ban saw it as the precursor of wider restrictions on gun ownership and largely ineffective because the features that qualified a gun as an "assault weapon" were unrelated to a how dangerous a weapon is. Also, criminals are more likely to use smaller guns rather than assault weapons.
But ban supporters counter that the semi-automatic weapons covered by the ban are ideal for mass murders, because the weapons fire very rapidly and can hold a large magazine of bullets. They point to mass slaying from the 1984 McDonald's massacre in San Ysidro, Calif., in which 21 died, to the 1993 Branch Davidians raid in which four federal agents were murdered.