Two GOP governors have banded together and issued a call for Congress to end the two-decade-old standoff over immigration policy and allow states to sponsor immigrants who want to resettle in America.

In an op-ed article in the Washington Post, Utah’s governor, Spencer Cox, and the governor of Indiana, Eric Holcomb, said their states have hundreds of thousands of open jobs that cannot be filled by people moving in from other states. Their respective states “need immigrants who are ready to work and help build strong communities,” the pair said.

“Though border security is a national concern, and a nonnegotiable requirement of national security in a world with drug cartels and terrorists, we believe that states should be able to sponsor whatever immigrants serve the needs of their communities,” they said. “As it is, the standstill on immigration hobbles both parties and, more seriously, endangers America’s long-term well-being.”

A number of bills intended to fix America’s broken immigration system have been introduced in the House and Senate in recent weeks, most of them pandering to the extremes on both sides of the debate and few with any chance of making it beyond the politically fractured Congress.

Debate on the topic in the House has been particularly shrill. One House Republican, Tony Gonzales, suggested that representatives from both sides of the aisle are pandering to their respective bases with some of the measures.

“There’s a reason why we haven’t gotten significant border security done and why we haven’t seen significant immigration reform done,” Mr. Gonzales said in an interview with the Washington Examiner last week. “It is in the interest of many politicians to have this crisis continue to flare up.”

Efforts in the Senate to get at the root of the problem have been more measured. A group led by Senator Sinema, an independent of Arizona, and Senator Tillis, a Republican of North Carolina, mapped out a plan late last year that would combine the issue of border security — a perennial demand from Republicans — with immigration reform, but the measure stalled out before the end of the session. The group is pressing ahead in the current Congress with that framework as a starting point.

Ms. Sinema’s bill would offer a path to citizenship for some 2 million immigrants who arrived in the United States as children, the so-called Dreamers that Democrats insist on being a part of any reform efforts. In exchange, the measure would change how the asylum system is enforced at the border, possibly allowing for the detention of asylum-seeking immigrants while their cases are being heard and expanding the use of Title 42 to deport back to Mexico immigrants crossing the border illegally.

“We’re committed to finding lasting solutions to this crisis,” Ms. Sinema said during a roundtable at the border with other senators last month. Getting anything passed by the Senate through the House is likely to be the legislation’s biggest hurdle.

Congress has not managed to pass any significant immigration reform since it approved a plan in 1986 during the administration of President Reagan. Lawmakers tinkered around the edges through much of the 1990s, but since 2007 nothing has managed to escape Capitol Hill. The Pew Research Center has suggested that two-thirds of Americans want some sort of clear path to citizenship for immigrants along with better enforcement at the border, but Congress has remained hamstrung.

In their article Tuesday, Messrs. Cox and Holcomb said the shrill voices on both sides have poisoned the debate in Congress.

“Voices on one side call for an overly liberal national immigration policy that overlooks the toll on border states and encourages too many people to take dangerous journeys that can end in legal limbo,” they said. “Meanwhile, voices on the other side call immigrants a burden, against all the evidence from economics and history.”

The governors want Congress to offer some sort of mechanism for states to choose who is allowed to emigrate where and when, in much the same way employers and universities are allowed to do now. Limits on the numbers of visas available to particular states could be set by Congress, they said, as well as restrictions requiring the states to assume responsibility for the new citizens.

Delegating some sort of immigration authority to the states is not a new idea. In 2019, another Utah Republican, John Curtis, offered up the State-Sponsored Visa Pilot Program Act in the House, but none in the chamber agreed to co-sponsor the bill and it never came up for debate. Governors Cox and Holcomb said the paralysis needs to end.

“Rapidly declining birthrates and accelerating retirements across the United States mean that our states’ already wide job gaps will grow to crisis proportions without more” immigration, the governors argue. “Many of these jobs require high-level skills and entrepreneurship. But states are also awash in unfilled entry-level, low-skill roles — essential in agriculture, health care and the service industries.

“Immigration sponsorship would give states a dynamic means to attract new residents, both from a pool of new applicants from abroad and from the ranks of current asylum seekers,” they said.