Cheap labor and pretty faces are always in demand, and Members of Congress want to deliver.

With comprehensive immigration reform in the deep freeze, frustrated lawmakers are aiming to push through niche bills this year to ease immigration for a range of workers, from farm hands to fashion models.

Sponsors say this might be the only way immigration changes can cut through the Congressional gridlock and radioactive rhetoric.

Immigration has been caught in a vortex where any legislation that grants visas to illegal immigrants is labeled “amnesty” and opposed by most Republicans, while smaller bipartisan bills expanding visas backed by businesses are blocked by members of the Congressional Hispanic Caucus in retaliation.

“I’d be inclined to say deal with the most vulnerable or not at all,” Rep. Luis Gutierrez (D-Ill.) said.

But there are signs that at least some smaller bills might reach the president’s desk this year.

In the House, a bill extending a visa program for doctors working in rural communities will be on the floor’s suspension calendar this week. And a series of other niche bills could head to the floor in coming weeks, including the fashion model bill sponsored by Rep. Anthony Weiner (D-N.Y.). A House Democratic aide said the models would generate “millions” in economic activity.

“Some discrete parts and individual pieces of immigration will certainly move forward,” a House Democratic aide said. The aide said there is hope that the Congressional Hispanic Caucus will let some bills move forward rather than try and kill everything to make a point.

“You don’t shove the entire apple in your mouth. You take a bite at a time,” the aide said.

Other bills in the works — some of which tweak or augment existing programs — include green cards to immigrants who create 10 jobs and invest $1 million, and another that would give green cards to foreign students at American universities who get advanced degrees.

Other bills would remove a provision that caps one country from using more than 7 percent of employment visas, and roll over unused green cards to future years.

Most of the bills are co-sponsored by Republicans and seen as small enough to potentially fly under the radar.

In the Senate, Sens. Dianne Feinstein (D-Calif.) and Larry Craig (R-Idaho) added a scaled back “AgJOBS” legislation to the Iraq War supplemental in committee last week, providing up to 1.35 million visas to farm workers. A smaller provision extending temporary visas to returning seasonal workers sponsored by Sens. Barbara Mikulski (D-Md.) and Judd Gregg (R-N.H.) also made it onto the bill.

But Senate leaders could decide to pare down the package as it heads to the floor.

“It’s fair to say that Sen. Feinstein has been working to see that this bill is passed, looking to have a vote on every moving vehicle, and she’s been told she can’t,” said Scott Gerber, spokesman for Feinstein. “We don’t know what will happen on the floor this week.” He added that Feinstein would continue to push for a vote at every opportunity if it is stripped from the bill.

“What agriculture needs is a stable, reliable source of labor, so they know they are going to have labor when they need it — when they need to plant, when they need to harvest, when they need to pack,” he said.

The seasonal worker legislation, which affects industries such as tourism, seafood and landscaping, has been blocked in the House, where it became entangled with aborted Democratic efforts to pass a scaled-down immigration reform package.

Democrats put the kibosh on their broader legislation after Republicans made it clear that they would not support it.

Democratic leaders meanwhile have kept an enforcement-only bill sponsored by Reps. Health Shuler (D-N.C.) and Brian Bilbray (R-Calif.) from coming to the floor.