Boca Raton, Fla. — President Barack Obama discredited the Goldstone Report critical of the way the Israeli military fought a war against Hamas in Gaza in December 2008.

He supported a United Nations resolution Israel accepted regarding the Jewish state’s handling of the Gaza flotilla in 2010; he made a middle-of-the-night phone call last year to get Egyptian authorities to rescue Israeli Embassy employees trapped by an angry mob; he provided Israel with two loan guarantees totaling $5.78 billion; he gave Israeli products what amounts to most favored nation status in the face of calls for a boycott of Israeli goods; and he enhanced Israel’s missile defense.

Despite all of that, more than three years into his presidency there is still a lingering perception among some Jews — particularly single-issue, pro-Israel voters who tend to be Republican — that Obama is anti-Israel, say observers. It was evinced during last week’s Republican presidential primary in Florida, and Robert Watson, director of American studies at Lynn University in Boca Raton, Fla., said he also found it among Jews when he spoke recently in Chicago and Detroit.

The 2012 election, most analysts believe, will hinge on the state of the economy, but in the Jewish community the Israel issue can figure prominently, in the race both for money and votes. And Watson, a registered Democrat who stressed that he is an author, professor and nonpartisan analyst, said he believes that “for the first time in a long time, Republicans may make inroads [in the traditionally solid Jewish Democratic vote] despite what Obama has done for Israel.”

Watson said that should the Republican nominee get as much as 40 or 45 percent of the Jewish vote, “the Democrats could be in trouble.” Although the Jewish vote is small, it is significant in three key swing states: Florida, Ohio and Pennsylvania.

Should the Republican presidential nominee do that well, he would set a record. The top vote getter — Ronald Reagan — got 39 percent of the Jewish vote in 1980.

Some polling data might bear out his sentiments. A survey released this month by the Pew Research Center found that the percentage of Jews who identified as Republican last year rose by nearly 50 percent since 2008, growing from 20 percent to 29 percent.

“While the majority of Jews are still Democrats or lean toward the Democratic Party, Democrats’ dominance among this group has weakened since the last presidential election,” the researchers wrote. “In fact, Jews are the only religious group analyzed in which the percentage who identify themselves as Republican (as opposed to leaning toward the GOP) has risen significantly.”

But exit polls taken during last week’s Florida primary found that only 1 percent of the Republicans who voted were Jewish, compared with 3 percent in 2008. Watson said that might suggest that Jewish Republicans are disgusted with all of the candidates and decided to sit out the primary (exit polls found that 58 percent of Florida Republicans who voted were unhappy with the Republican field). He said if Jewish Republicans sit out the general election too, that would translate into a “no” vote for Obama.

But now the Democrats are pushing back.

They recently posted a seven-minute video on the Internet in which Obama is shown speaking about America’s strong ties to Israel. Such Israeli leaders as Prime Minister Benjamin Netanyahu and Defense Minister Ehud Barak echo his words.

Rep. Debbie Wasserman Schultz of Florida, chair of the Democratic National Committee, said she is convinced Obama “will again earn an overwhelming majority of the Jewish vote.” He captured 78 percent in 2008.

She said that if Jews were “leaving the Democratic Party in droves and shifting to the Republicans, you would expect to see Jews switching parties, which didn’t happen.”

Sid Dinerstein, chairman of the West Palm Beach Republican Party, agreed that there was no increase in Jewish Republican registrations before the Florida primary. But he insisted that Obama’s position that Israel should agree to peace with the Palestinians based on its pre-1967 borders, with agreed-upon land swaps, “is an issue that is killing” the Democrats.

“Imagine if Obama wanted to give away half of the Vatican to a group that hated Catholics,” he said. “This is a position that would mean splitting Jerusalem and giving up the Wailing Wall. Can you imagine?”

The position, in fact, has been the one adopted by successive Israeli governments since the Oslo Accords in the 1990s. But Obama’s insistence that Israel halt all settlement activity seems to have hurt him with some Jews, who feel he has put the peacemaking onus on Israel.

Wasserman Schultz said Obama has a “remarkable record on Israel” in which he has “made it clear that the only path [to peace with the Palestinians] is through direct bilateral negotiations.” She said she has found that among Jews for whom “Israel is paramount … [Obama] has a stellar record and we are making sure Jews know about that.”

She said the Obama video is being e-mailed and shown at various events.

To counter the belief that Obama has never been to Israel, the video begins by showing pictures of him at the Western Wall and laying a wreath at Yad Vashem, Israel’s Holocaust memorial.

“He’s been to Israel twice, the last time just one month before the election in 2008,” Wasserman Schultz said. “And he has made a commitment to going again as president.”

Nevertheless, Watson, who is not Jewish, said that when he speaks once or twice each week in front of Jewish audiences in Florida and occasionally elsewhere he hears a constant refrain that Obama is not good for Israel.

“When I ask how they can say that, invariably they say Obama put his feet up and watched TV upstairs in the White House [dining with his family] while Netanyahu sat downstairs by himself, snubbed,” he said. “When I say that didn’t happen, they say, ‘No, it did.’ And if you try to explain to them otherwise, they don’t want to hear it. It has become an urban legend.”

The White House issued a statement denying the so-called snubbing incident, and even pointed out that Obama’s wife and daughters were in New York City at the time of Netanyahu’s visit.

Gerald Steinberg, a political science professor at Bar-Ilan University, said the false report was the creation of a media that likes to “over-dramatize disagreements.”

“It was an easy rumor to expand on,” he said. “That kind of hostility is part of the daily coverage here. Israelis don’t believe it and never did.”

Steinberg said the “attempt to poison the relationship” between the leaders of Israel and the U.S. goes back to 1977 when the Likud Party led by Menachem Begin won the election. The defeated Labor Party told the Carter administration “that it was a fluke and would not last long.” Some in the Carter administration believed it.

“A somewhat similar situation occurred during Obama’s first year and part of the second year,” he said. “There were Israelis on the left who thought they could force a change by a creating a crisis between Bibi and Obama … [in order to have] Tzipi Livni replace Bibi.”

Steinberg said Obama enjoys widespread popularity in Israel.

“Among the Israeli public he is much more popular than he is among Jews in the U.S.,” he said.

The American Jewish Committee’s annual survey of Jewish opinion released in September found that 48 percent of Jews disapproved of Obama’s job performance and only 46 percent approved. A year ago, Obama had a 57 percent approval rating.

Wasserman Schultz said the Republicans are “distorting and lying” regarding Obama’s relationship with Israel because “they know they can’t get anywhere on domestic issues. … Those are the issues we are in lockstep on — Medicare, Social Security, education, civil rights and civil liberties.”

She added that Republican Jewish groups are “putting love of party in front of love for Israel, and that is dangerous for Israel and leaves the impression that there is daylight between Israel and the U.S. That is wrong and it could be harmful.”