WASHINGTON (Reuters) - President George W. Bush’s demand for immunity for telephone companies that participated in his warrantless domestic spying program won an initial victory on Monday in the U.S. Senate.

President Bush speaks to reporters as Vice President Cheney looks on, in the Rose Garden, December 14, 2007. REUTERS/Kevin Lamarque

On a vote of 76-10, far more than the 60 needed, the Democratic-led Senate cleared a procedural hurdle and began considering a bill to increase congressional and judicial oversight of electronic surveillance of suspected terrorists.

It includes a provision to grant retroactive immunity to any telecommunications company that took part in Bush’s spying program -- surveillance without court warrants of e-mails and telephone calls of people in the United States -- begun shortly after the September 11 attacks.

Nearly 40 lawsuits have been filed accusing AT&T, Verizon and Sprint Nextel Corp. of violating U.S. privacy rights.

Backers of immunity, who include some Democrats as well many of Bush’s fellow Republicans, contend companies should be thanked, not punished, for helping defend the United States.

But civil liberties advocates and a number of Democratic lawmakers argue the courts should determine if any company violated privacy rights of law-abiding Americans.

Democrats vow to offer amendments in coming days to remove the immunity provision while backing a number of proposed new civil-liberty safeguards that enjoy broad support.

Sixty votes will likely be needed to prevail on any such immunity amendment in the 100-member Senate. “It’s going to be an uphill battle,” a Democratic aide said.

Sen. Chris Dodd, a Connecticut Democrat, interrupted his long-shot presidential campaign to return to Washington to help lead the charge against immunity. “For the last six years, our largest telecommunication companies have been spying on their own American customers,” Dodd said.

“That decision betrayed million of customers’ trust,” Dodd added. “But was it illegal? I don’t know. And if this bill passes in its current form, we will never know.”

The White House said in a statement, “Providing liability protection to these companies is a just result” and warned that allowing litigation “risks the disclosure of highly classified information regarding intelligence sources and methods.”

The House of Representatives last month defied Bush and refused to shield phone companies from lawsuits. Both chambers would have to agree to immunity before it could be granted.

The 1978 Foreign Intelligence Surveillance Act (FISA) requires the government receive the approval of a secret FISA court to conduct surveillance in the United States of suspected foreign enemy targets.

But shortly after the September 11 attacks, Bush authorized warrantless surveillance of communications between people in the United States and others overseas if one of the parties had suspected ties to terrorists.

Critics charged that Bush violated FISA, but he argued he had the war-time powers to do so. In January, Bush put the program under FISA’s authority. Terms remain secret.

In August, Congress bowed to Bush’s demands and expanded U.S. power to conduct surveillance without a court order.

The Senate bill would provide new protections of civil liberties, such as requiring tougher congressional and judicial oversight.