In recent weeks, some Democratic Party elected officials and strategists have grown increasingly anxious about the prospect that Mr. Sanders could capture the nomination. Many believe that they have a limited window to thwart any possible rise or streak of victories in the primaries, and that a wait-and-see approach — such as taking action after the Super Tuesday contests in early March — could end up being too late and too divisive for the party.

While there are informal discussions in Democratic circles about starting an organized effort to stop Mr. Sanders, so far no group of donors or organizations has stepped up to helm a broader campaign. On Tuesday, Third Way, a centrist think tank, sent an email to several hundred of its supporters in Iowa, encouraging them to highlight positions taken by Mr. Sanders that the group argues “will repel swing voters in battleground states.”

Mark Mellman, the president of Democratic Majority for Israel and a longtime Democratic pollster, said his group began developing an ad several weeks ago, as Mr. Sanders rebounded from a slide in the polls last fall. The group did not work in conjunction with any other organization.

The concerns about Mr. Sanders are two-pronged. The group and its allies worry that his questioning of the Democratic Party’s longstanding support for Israel could damage the United States’ historically strong relationship with the country.

If elected, Mr. Sanders would be the first Jewish president. Mr. Sanders, who doesn’t often speak about his Jewish upbringing, has called for an American policy toward Israel that addresses both Israeli security and a “pro-Palestinian” perspective. He spent time working on a kibbutz in Israel as a young man and opposes the movement, supported by some liberals, to boycott, divest from and sanction the country.

Mr. Mellman said his organization was equally concerned about the ability of the liberal septuagenarian to beat Mr. Trump, saying Mr. Sanders was in a “uniquely bad position” to win a general election.

“For many months people were saying he really didn’t have much of a chance, but you have to look at the data now and say he does have a realistic chance of winning Iowa and potentially the nomination,” he said. “This is the point at which there could be room for someone else to make their case and have a better candidate emerge.”