Massachusetts Sens. John Kerry and Edward Kennedy, along with a small number of other Senate Democrats, have threatened a filibuster to block the vote for Judge Samuel Alito's confirmation to the U.S. Supreme Court, FOX News has learned.

"Judge Alito has consistently made it harder for Americans to have their day in court. He routinely defers to the power of the government, no matter how extreme. And he doesn’t believe women have a right to privacy that’s protected by the Constitution," Kerry said in a statement.

"The president has every right to nominate Samuel Alito to the Supreme Court. It’s our right and our responsibility to oppose him vigorously and to fight against this radical upending of the Supreme Court," he added before announcing he would return to Washington early on Friday from Davos, Switzerland, where a Senate delegation was attending the World Economic Forum.

But a senior Democratic leadership aide told FOX News on Thursday that Senate Minority Leader Harry Reid and a majority of Democrats "do not support this action by Kerry and Kennedy. It is not politically advantageous for us."

The Senate's No. 2 Democrat said earlier in the day that while he opposes Alito, he thought it unlikely that Democrats would try to mount a filibuster.

"Having made a count, I have come to the conclusion that it is highly unlikely that a filibuster would succeed," said Senate Minority Whip Dick Durbin, D-Ill.

"There's been adequate time for people to debate," Reid added.

But on the Senate floor late Thursday, Kennedy said support is growing for a filibuster, and Durbin, along with Sens. Debbie Stabenow of Michigan and Paul Sarbanes of Maryland would back an attempt.

Republicans, however, have already said they'll have the 60 votes needed in the Senate to break a filibuster and end the debate, so such a threat probably won't have an impact.

"When you have the number of senators who have stated their intention to vote for cloture plus the number of senators who stated their intentions to vote for Judge Alito, you come to 60 or more," said Senate Judiciary Committee Chairman Arlen Specter, R-Pa.

A cloture vote, which would end a potential filibuster, is scheduled for Monday at 4:30 p.m. EST; if cloture is approved, the final vote would be scheduled for Tuesday at 11 a.m EST.

"It is time to establish an end point" in the debate over President Bush's selection, Frist, R-Tenn., said.

If confirmed, Alito will be replacing retiring Justice Sandra Day O'Connor. His swearing-in could come within an hour of confirmation.

Earlier Thursday, Alito picked up two new Democratic supporters, Sen. Robert Byrd of West Virginia and Sen. Tim Johnson of South Dakota. They join Sen. Ben Nelson of Nebraska, along with 52 Republicans, who have so far publicly pledged to vote for the conservative judge.

"His obvious intelligence, his obvious sincerity lead me to believe him to be an honorable man, a man who loves his country, loves his Constitution and man who will give of his best," Byrd said.

FOX News' Trish Turner, Brian Wilson and Catherine Donaldson-Evans contributed to this report.