Several Republican governors are urging GOP congressional leaders to stand firm next week in opposing legislation funding the Department of Homeland Security if it does not also overturn President Barack Obama’s executive action on immigration.

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Governors in both parties meeting in Washington this weekend warned of economic and security concerns should Congress fail to resolve its latest budget standoff. The agency’s $40bn budget runs out on 27 February, giving federal lawmakers only a few days to reach an agreement once they return from recess next week.

Homeland Security funding “is very important to not only our nation, but to our individual states because of our military installations – especially at a time that there are so many threats around the world with Isis and other terrorist groups,” said Oklahoma governor Mary Fallin, a Republican. “We hope that Congress will be able to find a resolution to get DHS funding passed, but we also know there are concerns and questions about immigration and the president’s powers.”

While some call for compromise, a handful of high-profile Republicans charged that stopping what they see as Obama’s unconstitutional power grab may be as important as resolving the funding dispute. The Obama administration put on hold plans to shield more than four million immigrants living in the country illegally from deportation after a federal judge in Texas temporarily blocked the effort the day the program was scheduled to begin.

Twenty-six states, led by Texas, filed suit in December arguing that the president does not have the authority to allow the groups of immigrants to legally stay and work in the US. The White House is appealing the ruling.

Indiana governor Mike Pence, who is weighing a Republican presidential bid, said the court ruling “vindicates the efforts of the Congress to use the power of the purse to prevent the administration from doing what the constitution does not permit them to do”.

Pence said he was “strongly urging” the Senate to support a House-passed bill that funds the Department of Homeland Security but also undoes Obama’s immigration actions.

Another Republican White House prospect, Wisconsin governor Scott Walker, suggested that while compromise might be possible, Republicans “need to be aggressive in pushing back on” the president’s immigration plans.

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“I think there’s a tremendous opportunity, particularly in light of the win that we had in court the other day, to make a further stand here about the president’s actions being an overstep on immigration,” he said, noting that Wisconsin is among the 26 states involve in the lawsuit. He said he hoped Congress would come up with a solution that “funds Homeland Security, but acknowledges the overstep of this president”.

Utah’s Gary Herbert said Republicans in Congress “have to get their leverage where they can”.

But like others, he was frustrated with congressional inaction on immigration despite repeated pleas from state leaders and business and immigrant groups to address the issue. “The solution is Congress: get off your fanny and get something done,” he said.

The stakes are high for millions of immigrants in the country illegally and the political fortunes of both parties heading into a presidential election.

“We’re talking about people’s pay checks. We’re talking about people’s lives,” Colorado governor John Hickenlooper, a Democrat who also serves as chairman of the National Governors Association, said of a budget impasse. “It is going to affect our states, and it’s going to affect every state in the country.”

Congress has little time to resolve the dispute before the Homeland Security budget runs out on Friday.

A department shutdown would have a limited impact on national security. Most workers across agencies, including the Secret Service, Transportation Security Administration and Customs and Border Protection, fall into exempted categories of workers who perform work considered necessary to protect human life and property and would stay on the job in a shutdown.

Most workers would not get paid until the shutdown ends, however. And all personnel involved in administering grants would be furloughed, including Federal Emergency Management Agency (Fema) workers who make grants to state and local governments, fire departments and others to help them prepare for or respond to various threats and emergencies.

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One possibility is a short-term extension of current funding levels, but House Speaker John Boehner said last weekend the House had done its job and he would “certainly” let a shutdown occur if the Senate did not act. Democrats have resisted any Homeland Security funding bill linked to the immigration order.

Like other Democrats, Delaware governor Jack Markell said the Republican-led Congress should simply fund the Department of Homeland Security without playing politics with immigration.

“It’s just like one more reflection of the dysfunction and the fact that the Republicans came down here and said that they were going to govern differently,” Markell said. “And you’re back to where it was.”

Tennessee’s Bill Haslam, who leads the Republican Governors Association, likened Obama’s executive order to “throwing a hand grenade” into efforts to adopt meaningful immigration reform.

“I think there’s a chance in this country to come to a real workable solution. I think there’s enough people understanding that, ‘OK, we’re not going to deport 12 million people,’” Haslam said. “But citizens of all types understand there’s a way to do government the right way.”