The House could introduce a new immigration bill, or several bills, to vote on. But given how little time they have, it's more likely they'd vote on a bill that's already been introduced.
Several immigration bills were passed out of House committees in the summer of 2013, but none of them have been brought to the House floor yet. If Speaker Boehner and House leadership want to take a vote on immigration in mid-June of 2014, these bills would be the only options that wouldn't require them to rush through the committee process first.
The Border Security "RESULTS" Act: requires the Department of Homeland Security to come up with a plan to meet security metrics at the border. The bill doesn't give DHS any money to do this, since its purpose is to get DHS to do more with what it has. This might make it less-than-attractive to border hawks who want more fences and drones. It's also been criticized by the Heritage Foundation, which sees it as a potential "back door to amnesty." However, the bill passed unanimously out of the House Homeland Security Committee — making it the only bipartisan immigration bill that has gone through the committee process.
The SAFE Act: the only bill that's gone through committee that deals with unauthorized immigrants — but deals with them punitively. It makes being in the country without papers a federal crime, and forces state and local law enforcement to participate in enforcing immigration laws. The SAFE Act has attracted a great deal of opposition from immigrant-rights advocates.
The AG Act: creates a new visa for farm workers, but requires them to return to their home countries every year to collect some of their earnings.
The SKILLS Act: increases visas for high-skilled workers, while cutting family-based immigration.
The Legal Workforce Act: requires all employers to use the government's electronic E-Verify system to ensure all of their workers are in the country legally. The Senate's comprehensive immigration bill also makes E-Verify mandatory, but does so in the context of a comprehensive reform that includes a path for the 8 million unauthorized workers currently in the country to start working legally.
It's unclear how much confidence House leadership has in these particular bills. None of them has been touched for almost a year. House leadership repeatedly declined to commit to bring them to the floor. Furthermore, they predate the "principles" that House Republican leadership released in January of this year, which were supposed to guide their approach to immigration reform.
Two other bills have been introduced that have also attracted the interest of both Democrats and Republicans, but haven't yet gone through the committee process:
The ENLIST Act: provides legal status and a path to citizenship for unauthorized immigrants who serve in the military. The bill was introduced by Republican Jeff Denham, and has 25 Republican cosponsors. But when Denham wanted to add it as an amendment to an authorization bill in May, House leadership shot it down, saying it wasn't the right venue for the bill.
H.R. 15: a counterpart to the immigration bill passed by the Senate, which was introduced by House Democrats in October 2013. H.R. 15 is largely identical to the Senate bill, but replaces the Senate bill's border security section (including the "border surge") with the "Border Security RESULTS Act" that passed the House Homeland Security Committee in 2013. The bill has 199 cosponsors, including 3 Republicans, but House leadership has repeatedly said they will not bring the bill to the floor.
