WASHINGTON — The Senate Judiciary Committee on Thursday approved a bill that would strengthen privacy protection for e-mails by requiring law enforcement officials to obtain a warrant from a judge in most cases before gaining access to messages in individual accounts stored electronically.

The bill is not expected to make it through Congress this year and will be the subject of negotiations next year with the Republican-led House. But the Senate panel’s approval was a first step toward an overhaul of a 1986 law that governs e-mail access and that is widely seen as outdated.

Senator Patrick Leahy, the Vermont Democrat who is chairman of the committee, was an architect of the 1986 law and is leading the effort to remake it. He said at the meeting on Thursday that e-mails stored by third parties should receive the same protection as papers stored in a filing cabinet in an individual’s house.

“Like many Americans, I am concerned about the growing and unwelcome intrusions into our private lives in cyberspace,” Mr. Leahy said. “I also understand that we must update our digital privacy laws to keep pace with the rapid advances in technology.”