WASHINGTON—Two bipartisan bills in the Senate would increase the number of visas and green cards available to high-tech workers, as lawmakers in both houses of Congress begin to sort out what is politically doable on the contentious issue of immigration.

The Senate bills are an effort to alleviate a pent-up demand for visas, particularly among technology companies that say they are losing good workers who are educated in the U.S. and then forced to leave.

The first bill, introduced Tuesday by Sens. Orrin Hatch (R., Utah), Amy Klobuchar (D., Minn.) and four others, would increase the number of high-skilled visas available each year to as many as 195,000 from the current cap of 65,000 and otherwise allow for more legal immigration to the U.S. They call the measure the Immigration Innovation Act, or I-Squared Act.

A second measure, backed by Sens. Jerry Moran (R., Kan.), Mark Warner (D., Va.) and four others would create an “entrepreneur’s visa” to allow people who want to start companies to stay in the country. Both measures would create new visa or green-card opportunities for foreign students who graduate from U.S. schools with advanced degrees in science, technology, engineering or math—something President Barack Obama supports as well.

“There’s broad based support for the concepts that are included,” Sen. Moran said in an interview. “I think there’s a strong opportunity for us to have success in the new Congress.”