WASHINGTON — When Rep. Shelley Berkley pitched her bid for the U.S. Senate to pro-Israel donors, the Nevada Democrat reportedly told them it came down to math.

In the U.S. House of Representatives, the leading pro-Israel lawmaker said, she was one of 435. In the Senate she’d be one of 100.

Now backers of the Jewish congresswoman are nervously crunching different figures: Her odds of winning in the wake of the formal launch of a House ethics inquiry.

The investigation launched July 9, and backed by Democrats and Republicans on the committee, will focus on allegations that Berkley’s championing of kidney care benefited her husband, Larry Lehrner, a leading kidney specialist in Nevada.

The inquiry arises from revelations in a New York Times report last year which found, among other perceived conflicts, that her successful efforts to block a federal bid to close a kidney transplant center saved part of a practice co-owned by her husband.

Berkley’s office said she was “pleased with the committee’s decision to conduct a full and fair investigation, which will ensure all the facts are reviewed.”

Those close to Berkley say the campaign dreads the fodder this gives Berkley’s opponent, Sen. Dean Heller (R-Nev.), who can now run ads beginning, as one insider put it, “Shelley Berkley, comma, under an ethics investigation, comma.”

Republicans have yet to mount ads citing the inquiry, but the National Republican Senatorial Committee made clear the day it was announced that the probe would be an issue.

“Nevadans deserve someone in the Senate who they can trust to work on their behalf and not someone — like Ms. Berkley — who puts her own financial and political interests first,” the GOP group said.

The inquiry is not likely to be resolved before the November election, although Berkley has said she hopes for a speedy resolution. Heller had been appointed to the Senate in 2011 after Sen. John Ensign, also a Republican, resigned in a scandal over alleged favors for a former lover and her husband. Polls before news of the ethics inquiry had Heller and Berkley in a virtual dead heat.

Pro-Israel groups say the loss of Berkley, a stalwart on the cusp of her possible elevation to the Senate, would be a blow.

“There are many members of Congress who support Israel because they think it is the right thing to do,” said Morton Klein, the president of the Zionist Organization of America, a group that often has hosted Berkley. “But few feel their commitment to Israel from the heart as Shelley Berkley does. She does not simply do what’s good for U.S-Israel relations, she does it with extraordinary sincerity and deep commitment.”

Berkley has faced the charges head on, saying that according to congressional ethics standards, benefits to constituents mitigate personal benefits for lawmakers. In her first ad after the inquiry’s launch, Berkley alluded to the charges in a pivot against Heller’s vote to restrain funding for Medicare.

“The Las Vegas Sun says Berkley’s ‘advocacy wasn’t driven for personal gain, it was aimed at helping Nevadans,’ ” the ad says. “And Dean Heller? Heller voted twice to end Medicare as we know it.”

Ben Chouake, who heads NORPAC, a leading pro-Israel political action committee backing Berkley, said she never hid her husband’s interest in keeping the kidney transplant center open.

“She did her job,” Chouake said, noting that the entire state delegation — Republican and Democratic, and including Heller — backed keeping the transplant center in state. “She’s not hiding the fact that her husband is a nephrologist. She’s always saying, ‘My husband Larry, the nephrologist.”

Berkley is a star in the Las Vegas Jewish community, according to Elliott Karp, the director of the city’s Jewish federation.

“She is clearly viewed as someone the Jewish community is proud of because she’s a great representative of the Jewish community in Nevada,” he said.

Berkley, who in appearances notes her Sephardic-Ashkenazi heritage with pride, is known affectionately among Democrats as the “Hadassah lady.” She was the only lawmaker who during a congressional salute to the group on its 100th anniversary in 2002 congratulated “my Hadassah sisters” from the House floor. She also consistently expresses pride in her pre-congressional involvement with the American Israel Public Affairs Committee.

With her oversize handbags, slightly stooped gait, steely coif, wide smile and high-volume intensity, Berkley, 61, has mastered conveying to her interlocutors the intense interest of an older relative.

Berkley’s pro-Israel impact is felt less through legislation — few of her pro-Israel initiatives make it to the House floor for a full vote — and more in how her rhetoric helps set pro-Israel parameters in Congress.

“If I’m putting something together, she’s the first person I call,” said Rep. Eliot Engel (D-N.Y.), who has co-sponsored a host of pro-Israel initiatives with Berkley.

In letters in recent months, Berkley and Engel have pressed the Obama administration to review the U.S. relationship with Turkey in the wake of its distancing from Israel and protested to foreign diplomats UNESCO’s designation of the Church of the Nativity as a world heritage site, describing it as a political maneuver by the Palestinians.

Berkley’s voice would be magnified in the Senate, Chouake said.

Sen. Joe Lieberman (I-Conn.), the Senate’s most pronounced pro-Israel voice but who is retiring this year, is headlining a fundraiser for Berkley in New York later this month.

Berkley’s outspokenness has earned her admirers on both sides of the political spectrum. Right-wing evangelicals adore her for bringing them Democratic credibility — she routinely appears at Christians United for Israel events. She has criticized the Obama administration for not consistently endorsing her view that the Palestinians are primarily to blame for the collapse in peacemaking.

Liberals revel in her long-running feud with Sheldon Adelson, the casino magnate and a major donor to Republicans, dating back to when he fired her as his legislative director in 1997 over differences on how to deal with unions. Berkley in a 2008 New Yorker profile of Adelson accused him of abusing the “raw power of money.”

Berkley lashed out at the administration of George W. Bush for making Israel more vulnerable as a result of the Iraq War.

“The only nation that seems to have benefited by our invasion of Iraq is Iran, which is a far greater threat to Israel than Iraq was,” she told JTA in October 2004, accusing then-Vice President Dick Cheney of “deceiving” her and others in the pro-Israel community when he made the case to them for the Iraq invasion.

“She’s not afraid of repercussions, no matter who the president is,” Engel said.

M.J. Rosenberg, a critic of Israeli policies from the left, said he tends to hold his fire when it comes to Berkley because she is unafraid to break with her allies among pro-Israel hawks.

“She speaks out against anti-Muslim or anti-Arab bigotry,” Rosenberg said. “She is a strong backer of working people, unions and minorities. I totally disagree with her position on the occupation, but she is a liberal and she is certainly not a hater.”

Berkley has outraised Heller this cycle $6.2 million to $5.8 million, and her campaign and donors show no signs of flagging. Nevada is seen a swing state and critical to maintaining Democratic control of the Senate.

Talking Points Memo, the liberal news site, last week reported that the Democratic Senatorial Campaign Committee pressed ahead with plans to reserve $2.3 million in TV ad spending for Berkley even after the announcement of the ethics inquiry.

Still, the House probe could hurt Berkley, who is well known in and around Las Vegas but had planned on using the coming months to introduce herself statewide.

“She is from a very small urban district, and not well known outside, said Jon Ralston, a Las Vegas Sun reporter and the doyen of the state’s political reporters. “This is not a good way for her to be introduced to the voters of Reno. It’s going to damage her; it’s a question of how much.”