Asia has passed Latin America in recent years as the largest source of immigrants to the United States. “On the national level, at least on the presidential, you’ve seen throughout the years more and more support for the Democratic Party,” said Christine Chen, the executive director of the nonpartisan group Asian and Pacific Islander American Vote. But “even a month before the election, 30 percent of our electorate that was polled was actually undecided.”

Shawn Steel, a Republican committeeman from California who has pressed his party to reach out to Asian-American voters, said that many Asians were trapped in a bureaucratic backlog even though “they followed the rules, they paid the fees, they had the interviews.”

“Largely because of the inefficient immigration policy, they are being denied an expedited hearing,” Mr. Steel said. He argued that an “aggressive amnesty” would give a “special favoritism to those that did not wait in line,” which would be unfair to many Asians.

The broad bipartisan bill that passed the Senate in June offers major benefits for Asian immigrants but also includes provisions that could reduce future immigration from some Asian countries.

The bill would eliminate the cap on the number of residency visas, known as green cards, available each year for spouses and children of legal permanent residents, a group that includes many Asian immigrants. It would also eliminate within 10 years the current backlog of more than four million foreigners who applied through legal channels and have been waiting, in many cases for a decade or more, for green cards. Under those terms, foreigners who had been waiting the longest for family and employment green cards, including many Asians, could receive their visas quickly.

But the bill would also eliminate categories of green cards for siblings of United States citizens and for married sons or daughters of citizens who are over 31 years old. Many Asian immigrants have applied through those categories, and Asian-American groups opposed those provisions.

The young man at the immigration event was not the only one asking Mr. Obama to make changes unilaterally. At the first of two events for Democratic Party contributors in San Francisco, a man repeatedly yelled “executive order” as the president discussed impediments to his agenda.