The Hill newspaper said legislation by Senate Democrats won 51 votes; a Republican alternative won 38.
Either plan needed at least 60 votes to advance, the newspaper said.
In a procedural move that would allow the bill to move to the floor in the future, Senate Majority Leader Harry Reid, D-Nev., switched his vote to "no."
Sen. Mark Warner of Virginia was the only Democrat who voted for the Republican legislation. Sen. John McCain, R-Ariz., was among the eight Republicans who voted against the GOP bill, The Hill reported.
The newspaper said the Congressional Budget Office estimates the cuts could cost 750,000 jobs this year.
However, President Obama is scheduled to meet with congressional leaders from both parties Friday at the White House to discuss the sequester.
None of the participants expects the White House meeting to produce a breakthrough, The Wall Street Journal reported.
The cuts -- which will run through the end of the fiscal year in September unless lawmakers intervene -- are the first of a decade-long plan to cut spending $1.2 trillion for nearly every federal program, except for military personnel and entitlement programs such as Medicare and Social Security.
The cuts were included in the 2011 deal to raise the federal debt limit. They represent between 2 percent and 3 percent of the federal government's annual $3.5 trillion budget.
Senate Democrats offered an alternative bill that would pay for the sequester through the end of the fiscal year with a combination of a minimum 30 percent tax on millionaires and cuts to defense and farm payments.
"Now, less than 48 hours before the clock runs out, all they've offered is a gimmicky tax hike that's designed to fail," Senate Minority Leader Mitch McConnell, R-Ky., said on Capitol Hill. "I hope they're not expecting a round of applause for this particular act of political bravery."
Senate Republicans had an alternative bill that would have the same $85 billion in budget cuts but put the onus on Obama to determine which programs would be cut.
The White House has said Obama will not support any sequester alternative that doesn't include a "balanced plan" that includes domestic program cuts and additional tax revenue collected from some corporations and high-income people.
The Republican-controlled House has not offered its own version of an alternative to the sequester cuts.
Speaker John Boehner, R-Ohio, has said his chamber will only consider an option that has passed the Senate.
The House passed two sequester alternatives near the end of last year's Congress that didn't become law.
"We should not have to move a third bill before the Senate gets off their ass and begins to do something," Boehner told reporters Tuesday.
Notable deaths of 2014 [PHOTOS]