As one bipartisan group of senators released its blueprint on Monday for a comprehensive overhaul of the immigration laws, another group in the Senate was ready to present a proposal addressing one dysfunctional aspect of the system: a shortage of visas for highly skilled immigrants working in science and technology fields.

Four senators, led by Orrin G. Hatch, a Republican from Utah, will introduce a bill on Tuesday that would greatly increase the number of temporary visas available for those immigrants, and would also free up permanent resident visas, known as green cards, so more of those immigrants could settle in the United States and eventually become citizens.

The bill will be the first legislation introduced in Congress on immigration in a year when the once-toxic issue has gathered surprising political momentum. Lawmakers who have shied away from it in recent years are now offering proposals that they are framing as practical solutions to fix a failing system.

Major technology employers like Microsoft, Oracle and others have been calling for years for more visas for foreigners with computer, engineering and mathematics skills, saying they have more jobs than they can fill with Americans who are graduating in those fields from American universities. White House officials and lawmakers on both sides of the aisle acknowledge that a broad overhaul, including a pathway to citizenship for 11 million illegal immigrants, would have a better chance at attracting votes for passage, especially among Republicans, if it had vigorous support from business.