SAN FRANCISCO — Kleiner Perkins Caufield & Byers, the U.S. venture capital firm that provided startup financing to Inc. and Google Inc., lost its second bid to force partner Ellen Pao to move her sex-discrimination claims to arbitration rather than proceed with a state lawsuit.

California Judge Harold Kahn in San Francisco said in a hearing today that he didn’t agree with Kleiner’s position that Pao’s claims had to be sent to arbitration because they are connected to agreements for the limited liability companies that manage Kleiner’s investment funds, which contain arbitration clauses.

“KPCB is disappointed in Judge Kahn’s decision and intends to file an appeal believing it has strong arguments and precedent to move the matter to arbitration,” Kleiner said in an e-mailed statement. “Ms. Pao, like other partners, signed a variety of standard agreements and it is these agreements with the managing LLCs that govern her claims and require, among other things, that disputes be resolved through arbitration.”

On July 10, Kahn rejected the firm’s argument that it had an agreement with Pao, an employee there since 2005, requiring the case to be resolved out of court. Kahn found that no such agreement existed, although he allowed Kleiner to refile its request.

Alan Exelrod, Pao’s attorney, said he was pleased with the judge’s decision in an interview after the hearing.

Kleiner, Perkins, Caulfied & Byers, with a roster of senior partners including Al Gore and Colin Powell, is thus still embroiled in a lawsuit that’s the talk of the valley.

The legal fight offers an inside peek at the firm’s jet-setting ways, internecine fights over board of director seats at “portfolio” companies and the cost of social snubs.

Pao, a 42-year-old Harvard Law School graduate, joined the firm in 2005.

Pao accuses Kleiner of blocking her and other women from advancement and lucrative positions she claims are reserved for men. The firm has proclaimed its innocence and alleges Pao “twisted facts and events in an attempt to create legal claims where none exists.”

Pao still works at the firm, which calls itself one of the most “progressive” companies in Silicon Valley in its hiring practices. Kleiner says that one-quarter of its senior partners are women.

Since Pao filed the lawsuit in May in San Francisco Superior Court, the tech world has been abuzz. Tech web sites have closely covered the case, even live blogging routine hearings.

In larger context, Pao’s suit highlights an oft-heard claim that, despite progress, it’s difficult for women to break through in Silicon Valley’s male-dominated culture. When Google executive Marissa Mayer was named Monday as the new CEO of Yahoo, the story was widely depicted as a testing of the valley’s renowned glass ceiling.

“This is a garden variety sex harassment case,” said Stanford University law professor Deborah Rhode, a sexual harassment legal expert. “It got the buzz because this is such a male dominated culture and it’s such a large firm” with a good reputation.

Kleiner opened its doors in Menlo Park in 1972, when few independent investment firms existed. Its rise mirrored that of Silicon Valley’s. The firm’s early investments in some of the computer industry’s biggest names propelled it to prominence. There were few if any women at the top of the firm’s organization chart for most of its history, a reflection of the all-male culture that was prevalent when Silicon Valley emerged.

Kleiner has hired high-priced employment attorney Lynn Hermle to do battle with Pao’s equally high-profile attorney, Alan Exelrod, who in 1998 won a $7 million sexual harassment judgment from what then was the world’s largest law firm, Baker & McKenzie.

Hermle’s first line of defense is an attempt to move the lawsuit from the public glare of open court to the closed doors of arbitration, where private judges render confidential decisions. Judge Harold Kahn will hear Kleiner’s arguments for arbitration on Friday.

Regardless of what happens next, damage already is done.

“The last several days have been a difficult time for me and my partners at Kleiner Perkins Caufield & Byers, a firm I’m proud to have been a part of for 32 years,” senior partner John Doerr said in a recent Web message to the firm’s employees. “It is not easy to stand by as false allegations are asserted against the firm, especially because legal constraints prevent us from responding fully at this time.”

The firm’s senior partners and lawyers are refusing to discuss the case with the media, referring reporters to its publicly accessible court filings. Similarly, Pao and her attorney claim they aren’t seeking media attention and are also declining to comment.

But the court records are full of accusations, counter accusations and long-simmering resentments of Pao and her bosses at Kleiner, who labeled her a poor performer passed over for promotion only for lack of merit.

Pao earned an undergraduate engineering degree from Princeton University and a law degree from Harvard. While working for a prestigious law firm, she earned a master’s of business administration from Harvard and ultimately went to work in June 2005 as Doerr’s chief of staff at Kleiner. She said she was called a “junior partner” and was told she would move into a full-time investing role in three years if she did a good job.

She claims her troubles at the firm started almost upon her arrival.

Pao said she filed her lawsuit after enduring five years of harassment from Ajit Nazre, a junior partner when she was hired who was later promoted to senior partner. Nazre left the firm in January and didn’t respond to email and phone messages.

Pao claims that senior partners such as Doerr ignored her complaints about Nazre. She concedes she and Nazre had “two or three” consensual sexual encounters but alleges he began harassing her after she broke off their relationship.

According to her lawsuit, Nazre began excluding her from important meetings, email chains and withholding necessary information for her to succeed at Kleiner. Instead of addressing her alleged problems, Pao claims partners either ignored her complaints, joined in her ostracism — or worse.

She said senior partner Ray Lane pressured her to drop her complaints and suggested that she should marry Nazre. She also claims another senior partner, Randy Komisar, sent her a sexually explicit book of poems and pictures on Valentine’s Day 2007 with a handwritten inscription.

The firm said Pao never complained about Nazre or any other harassment until she hired her attorney in late 2011. The firm claims that she even thanked Lane for supporting her during her breakup with Nazre, a “married peer.”

Pao also charges she was excluded from business-related outings that included a private jet ski trip to Vail, similar jet trips to New York and important dinner parties with influential executives hosted at a senior partner’s San Francisco residence. Pao alleges that the dinner snubs were particularly embarrassing because she lived in the same building and ran into attendees arriving for dinner. Pao said she was told the dinner affairs were attended exclusively by men because women “kill the buzz.”

Kleiner’s attorneys accuse Pao of misconstruing Komisar’s gift, which the company says was actually purchased by Komisar’s wife. The book at issue is Leonard Cohen’s “The Book of Longing,” which the songwriter wrote during his five-year stay at a Zen monastery. Komistar is a Buddhist.

Adding further intrigue, Pao’s husband, Alphonse “Buddy” Fletcher Jr., is involved in a well-publicized racial discrimination lawsuit in New York City with the Dakota Apartments — the place where John Lennon was killed — over his attempts to buy a fifth apartment. Fletcher is a well-known philanthropist who lived with his same-sex partner at the Dakota for 10 years until marrying Pao in 2007. They have a 3-year-old daughter.

Regardless of whether the case plays out publicly in court or behind closed doors in arbitration, longtime observers of the Silicon Valley scene are taken aback that the venerable firm is accused of such unseemly conduct.

Jamis MacNiven, owner of the iconic Silicon Valley watering hole Buck’s of Woodside that Kleiner partners have frequented for decades, said he was stunned to learn of the lawsuit because of the firm’s “impeccable reputation.”

“These are the most normal people in the world,” MacNiven said.

(The AP and Bloomberg contributed to this report.)