WASHINGTON — The so-called “kishkes issue” — what does President Obama, deep down, really feel about Israel — is now being addressed at the highest level by Obama himself.

Obama dropped in on a White House meeting Tuesday of lay and rabbinical leaders of the Conservative Judaism and Jack Lew, the president’s chief of staff. During his 20 minutes at the hourlong meeting, Obama emphasized his affection for Judaism and Israel, and like Vice President Joe Biden last week in a similar meeting with organizational leaders, his frustration with perceptions that he is cool toward the Jewish state.

The tone, coupled with blitzes of Jewish communities by Democratic leaders in recent months, reinforces the impression that the party’s leadership is unsettled by Republican inroads into what for decades has been a Democratic base constituency.

The presidential visit was “informal,” although the group of Jewish leaders knew a drop-by was likely. So when Obama walked into the Roosevelt Room, Rabbi Julie Schonfeld, the executive vice president of the Conservative movement’s Rabbinical Assembly, was ready with the traditional blessing for heads of state.

The account of what happened next is based on detailed notes by a person in attendance and confirmed by broader descriptions by others in attendance.

Obama opened by describing what he said was the “overlap” between his priorities and those of the Jewish community, both domestically and abroad. His first question was from Arnold Eisen, the chancellor of the Jewish Theological Seminary, who asked about the role of religious leaders in the public sphere.

Obama responded by speaking about the need to be part of a community, and he paraphrased a Talmudic injunction about Jews being responsible for one another, applying it to Americans.

Schonfeld then thanked the president for his efforts to revive the economy, as well as for isolating Iran and initiating U.N. Security Council condemnation of Syria’s bloody repression of regime opponents.

She then asked the president how best to convey hope to Americans.

Obama reiterated his thoughts about community and the obligations of Americans for one another, and described Republicans as seeing government as a negative power that unfairly distributes wealth from the hard-working to the lazy.

Rabbi Steven Wernick, the executive vice president of the United Synagogue of Conservative Judaism, then asked the president how the rabbis could push back against perceptions that he was hiding his true feelings about Israel.

“Jack always tells me I’ll get asked the kishkes question,” he said, referring to Lew, who is an Orthodox Jew, and using the Yiddish term for “guts.”

Obama said the question dated back to 2008 and for him was a bizarre reversal: Until then, he said, during his rise as a state politician in Illinois and then as U.S. senator, he had been depicted by some on the left as a “stooge” for Israel because of his close friendship with Jews and others in the pro-Israel community.

The president blamed several elements for the reversal: The reluctance among some Jews to credit someone with the middle name Hussein, and the son of a Muslim, with being pro-Israel; the quirk of history of a center-left government in the U.S. overlapping with a center-right government in Israel, and the resulting perception that Obama was pressing Prime Minister Benjamin Netanyahu too hard to shut down settlement expansion; and the fact that the Republicans seized these elements to advance a narrative that he was unfavorable to Israel.

Obama made clear that he resented the narrative, calling it “unfounded.” He said his support for Israel was evident not just during his administration through enhanced security assistance, but also during his days in the Senate.

Obama also complained that no one questions the pro-Israel bona fides of Mitt Romney, the all-but-certain Republican presidential nominee; Sen. Mitch McConnell (R-Ky.), the minority leader in the U.S. Senate; or Rep. John Boehner (R-Ohio), the speaker of the U.S. House of Representatives.

The administration’s pushback on this issue was evident as well in a May 21 meeting featuring top White House officials and some 70 Jewish community leaders. Biden took aside Morton Klein, the president of the Zionist Organization of America, and complained about a mutual acquaintance who was writing “scurrilous” emails about Obama’s record, according to Klein. Biden then repeated the complaint in his address to the whole group of Jewish leaders, according to Klein and another participant.

Matt Brooks, the director of the Republican Jewish Coalition, said that Obama’s complaints were disingenuous, noting that until this year a number of Democrats had complained about Obama’s approach to Israel, particularly his call to use the 1967 lines as the basis for negotiating the future borders of a Palestinian state.

“On the eve of his reelection, he’s engaging in a charm offensive with the Jewish community and has turned down some of the rhetoric,” Brooks told JTA. “But I don’t believe he has changed his fundamental policies in any regard.”

Obama in his talk with the Conservative leaders also said that he had read deeply about Judaism, saying he probably knew more about the topic than any other president. Brooks derided the assertion as “narcissistic.”

Those at the meeting said they were impressed by Obama’s remarks.

“He talked very passionately about his personal sense of commitment to the values that are reflected in the U.S.-Israel relationship and the feelings he shares with the American Jewish community for Israel,” Rabbi Jack Moline, the spiritual leader of the Agudas Achim Congregation in Alexandria, Va., and the Rabbinical Assembly’s director of public policy, told JTA.

Moline arranged the meeting between Lew and the Conservative leaders.

“My impression is that this was an effort by the White House to reach out to faith leaders of various communities, and we were just fortunate to be among the first,” Moline said.

In a blog post on the JTS website, Eisen said that Obama’s quest for Jewish approval was a positive.

“He clearly cares what the Jews of America think of him,” Eisen wrote. “This has to be a good thing for us and for Israel; I believe it is also a good thing for America.”

(Zach Silberman of the Washington Jewish Week contributed to this report.)