When the Republicans take control of Congress in January, they may act, with bipartisan support, to raise the H-1B cap.

It's going to be sticky, because the Republicans will be working on parallel paths. They will be taking President Barack Obama to task over his executive actions on immigration reform, as they pull together votes for a separate, stand-alone bill on the H-1B visa, according to several Capitol Hill sources who spoke only on background.

The groundwork for a bipartisan bill is in place. The top legislative candidate is the I-Squared Act, which was introduced in the Senate by two Republicans and two Democrats, Sens. Orrin Hatch (R-Utah), Amy Klobuchar (D-Minn.), Marco Rubio (R-Fla.) and Chris Coons (D-Del.). The bill had 26 co-sponsors.

The I-Squared bill, first introduced in the Senate in 2013, did not advance because of the Senate’s focus on comprehensive immigration reform. That bill is now seen as the leading legislative candidate in 2015 for raising the H-1B cap.

The Senate's Democratic leaders have traditionally opposed stand-alone H-1B bills because they didn't want to risk losing tech industry support for comprehensive immigration reform.

Just before the November election, Hatch, who become president pro tempore of the Senate, that body's second most powerful position, outlined the incoming Senate leadership's "innovation agenda." These are "priorities" that he plans to "help advance early in the next Congress." It includes support for the I-Squared bill.

The initial iteration of I-Squared called for raising the base H-1B cap from 65,000 to 115,000; another 20,000 H-1B visas are set aside for advanced degree graduates of U.S. schools. It also included mechanisms -- based on demand for the visa -- to allow the cap to rise as high as 300,000.

There is no certainty that these efforts to raise the H-1B cap will succeed, and it's possible that acrimony with the White House over immigration will derail action on an H-1B-specific bill.

Another factor in the mix might be former Florida governor Jeb Bush, who may be about the closest thing to an announced candidate for the Republican presidential nomination that his party has right now. If Congress takes up any immigration issue, he may speak out.

Bush, who recently announced on Facebook that he has "decided to actively explore the possibility of running for President of the United States," is a strong advocate for raising the H-1B cap, much like his brother, former President George W. Bush.

Gov. Bush detailed his position in Immigration Wars (Simon & Schuster, 2013), a book he co-authored with attorney Clint Bolick. He also sketched out his views as co-chair of the Council on Foreign Relations' task force on immigration policy.

Bush wrote that the 85,000 H-1B cap is "hopelessly inadequate to preserve America's leadership role in technology."

Any H-1B legislation must get through Sen. Charles Grassley (R-Iowa), who will be heading the Senate Judiciary Committee. Grassley has been a leading critic of the H-1B visa and believes it is used to displace U.S. workers.