"As many Americans continue to struggle without benefits, we respectfully request that the House immediately consider this bill or a similar measure," seven House Republicans wrote Boehner and House Majority Leader Eric Cantor on Monday. They released their letter as the Senate voted 59-38 for the bill.
The outlook remains cloudy, though, for legislation that took three months to move through the Senate.
In reply, a spokesman for Boehner said, "As the speaker said months ago, we are willing to look at extending emergency unemployment insurance as long as it includes provisions to help create more private sector jobs."
The spokesman, Michael Steel, said Senate Democratic leaders had last week "ruled out adding any job measures at all."
The White House-backed measure would retroactively restore benefits that were cut off in late December, and maintain them through the end of May. Officials say as many as 2.3 million jobless workers have gone without assistance since the law expired late last year. If renewed, the aid would total about $256 weekly, and in most cases go to men and women who have been off the job for longer than six months.
Sens. Jack Reed, D-R.I., and Dean Heller, R-Nev., the bill's leading supporters, said they were willing to consider changes in hopes of securing passage in a highly reluctant House.
Heller also said he was seeking a meeting with Boehner to discuss the measure, although the senator's office did not respond to a request for further information about any session with the speaker.
President Barack Obama, House Democrats and organized labor also seek a renewal of the expired benefit program for the long-term unemployed, but the bill's prospects likely reside with members of the Republican Party.
The Senate vote itself, seven months before congressional elections, capped a bruising three-month struggle. Fifty-one Democrats, two independents and six Republicans voted for approval.
The bill was the first major piece of legislation that Democrats sent to the floor of the Senate when Congress convened early in the year, the linchpin of a broader campaign-season agenda meant to showcase concern for men and women who are doing poorly in an era of economic disparity between rich and poor.
In the months since, the Democrats have alternately pummeled Republicans for holding up passage and made concessions in an effort to gain support from enough GOP lawmakers to overcome a filibuster. Chief among them was an agreement to pay the $9.6 billion cost of the five-month bill by making offsetting spending cuts elsewhere in the budget.
Six Republicans voted for the bill.
The measure's progress in the House is complicated by opposition from many conservatives who comprise the majority, as well as outside groups like the Club for Growth, which has urged lawmakers to oppose the bill.
Nor does the presence of only seven Republicans' signatures on the letter to Boehner impart much momentum.
They include Reps. Frank LoBiondo; Chris Smith and John Runyan of New Jersey; Michael Grimm, Peter King and Chris Gibson of New York, and Joseph Heck of Nevada.
Nevada has one of the highest unemployment rates in the country, 8.5 percent in February, according to the Bureau of Labor Statistics, and Heller played a persistent role in pushing the bill through the Senate.
Unlike Heller and Heck, though, Rep. Mark Amodei, the state's other Republican in Congress, has refrained from joining in the campaign to resurrect the expired program. Brian Baluta, a spokesman, said the state's unemployment rate has fallen since the program expired at the end of last year, the provisions in the bill to cover the costs are suspect, and the measure does not help create jobs.
The drive to renew the lapsed program comes as joblessness is slowly receding nationally, yet long-term unemployment is at or above pre-recession levels in much of the country. According to the Bureau of Labor Statistics, it accounts for an estimated one-third or more of all jobless individuals.
In a study last summer, the Urban Institute reported that "relative to currently employed workers, the long-term unemployed tend to be less educated and are more likely to be nonwhite, unmarried, disabled, impoverished and to have worked previously in the construction industry and construction occupations."