WASHINGTON—Congress is expected this week to pass yet another short-term spending bill to avoid a government shutdown, prompting a bipartisan pair of senators to introduce a narrow bill on immigration aimed at ending the impasse that has hindered a two-year budget deal.

The government’s current funding expires at 12:01 a.m. Friday. Immigration has become entangled in the spending negotiations since President Donald Trump last year ended an Obama-era program that protects young people who were brought to the U.S. illegally as children, known as Dreamers.

Senate Armed Services Committee Chairman John McCain (R., Ariz.) and Sen. Chris Coons (D., Del.) plan to unveil a proposal that offers a path to citizenship for Dreamers and orders a comprehensive study to determine what border-security measures are needed. But the bill stops well short of almost all of the president’s demands—including immediate funding for the wall along the southern border—and is likely to meet a chilly reception from conservative Republicans.

Still, Mr. McCain, who was diagnosed with brain cancer last summer and has been working from Arizona since late 2017, retains powerful sway among his GOP colleagues. His bill with Mr. Coons could also benefit from good timing, as the March 5 deadline draws closer and lawmakers grapple with the political consequences of failing to reach any agreement.

“It’s time we end the gridlock so we can quickly move on to completing a long-term budget agreement that provides our men and women in uniform the support they deserve,” Mr. McCain said in a statement Sunday.