Republican-backed measure would meet Trump’s wall demand while the second would extend funding for closed agencies

The Senate will vote on Thursday on a pair of bills that could end the month-long partial shutdown of the federal government – if passed.

The first bill, a Republican-backed measure, would meet Donald Trump’s demand for a $5.7bn wall along the southern border in exchange for temporary protections for young undocumented immigrants. The second would extend funding for the agencies that are currently closed through to 8 February.

The Senate majority leader, Mitch McConnell, and minority leader, Chuck Schumer, said on Tuesday that they would bring the duelling bills to the floor, in the first sign of progress after a 32-day stalemate over the partial shutdown, which has left as many as 800,000 government workers without pay.

They announced the compromise on the Senate floor on Tuesday, with Schumer predicting that the short-term funding proposal “could break us out of the morass we are in”.

“People are saying: isn’t there a way out of this mess? Isn’t there a way to relieve the burden on the 800,000 federal workers not getting paid? Isn’t there a way to get government services open first and debate what we should do for border security later?” he said. “Well, now there is a way.”

However, it is far from certain whether either bill can garner enough support to pass the chamber. Democrats, who are opposed to granting funding for a border wall, likely have the votes to block Trump’s proposal. The Democratic proposal would have to win the support of at least 13 Republicans to reach the 60-vote threshold.

The Senate approved a short-term funding bill in December that would have averted a shutdown and kept the agencies running until 8 February. That measure passed the Senate without any opposition but Trump later said he would not support it because the plan did not include funding for his wall.

The Republican-controlled House declined to vote on the measure. Since re-taking the majority in the House, Democrats have passed a number of bills that would reopen the shuttered agencies, but the Senate has refused to vote on them, arguing that they won’t take up legislation the president won’t sign.

A union representing FBI agents warned on Tuesday that the partial federal government shutdown has “hindered” the bureau’s ability to conduct operations and pursue investigations. Thousands of union members are among hundreds of thousands of federal workers and contractors now without pay for a fifth week.

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As the FBI Agents Association released a report containing firsthand accounts of how the shutdown has affected operations, its president, Tom O’Connor, demanded Congress and Donald Trump fully fund the FBI.

“The failure to fund the FBI undermines essential FBI operations, such as those designated to combat crimes against children, drug and gang crime and terrorism,” O’Connor told reporters.

He declined to say whether Americans were less safe as a result of the shutdown.

“I will leave that question up to you to answer,” he said.

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The union’s plea came as the shutdown continued to affect other government services across the country.

The Transportation Security Administration said the percentage of its airport screeners missing work hit 10% on Sunday, up from 3.1% on the comparable Sunday a year ago.

The screeners, who are without pay, have been citing financial hardship as the reason they cannot report to work. Even so, the agency said it screened 1.78 million passengers on Sunday with only 6.9% having to wait 15 minutes or longer to get through security.

In Washington, Senate Republicans released legislation designed to meet Trump’s offer to break the impasse and end the shutdown. The measure would include temporary relief from deportation for young undocumented immigrants known as Dreamers and other threatened groups, in exchange for meeting Trump’s demand for $5.7bn to build a southern border wall.

But Democrats have vowed opposition to Trump’s wall, calling it an impractical, “medieval solution” to a “21st-century problem”.

On Twitter, Trump wrote: “Never seen [McConnell] and Republicans so united on an issue as they are on the Humanitarian Crisis & Security on our Southern Border. If we create a Wall or Barrier which prevents Criminals and Drugs from flowing into our Country, Crime will go down by record numbers!”

The status of the annual State of the Union address remains unclear, one week after House speaker Nancy Pelosi requested Trump postpone it. On Tuesday, Fox News reported that the White House sent a letter to the House sergeant-at-arms to schedule a walk-through for the address, which is due to be held on 29 January.

White House spokesman Hogan Gidley said Trump could deliver the address from another venue if Pelosi blocks him from doing so in the House chamber.

“There are many ways he can deliver the State of the Union address,” Gidley said. “I’m not going to get ahead of anything he would announce.”

In his offer to Democrats on Saturday, Trump offered to extend temporary protections for Dreamers and those fleeing disaster zones. On Tuesday, the supreme court said it would not pick up a case concerning Trump’s previous attempt to end protection for Dreamers, meaning they will remain in limbo for the next few months.