Sen. Lindsey Graham of South Carolina on Sunday said a deal that combines $5 billion for President Trump’s border wall with an extension of legal status for “Dreamers” and people fleeing conflict or natural disasters back home could “save the day in the Senate” and reopen the government.

“We need to start talking again,” the Republican told CNN’s State of the Union. “To my Democratic friends: There will never be a deal without wall funding.”

He said in return, immigrants brought to the U.S. illegally as children should be eligible for legal status and authorization to work for three years under a bill he’s put forward with Sen. Richard Durbin, Illinois Democrat.

Persons living in the U.S. under Temporary Protected Status — for instance, persons fleeing civil war or hurricane wreckage — could also receive temporary, renewable status to stay in the country under the senator’s deal.

Mr. Graham said Democrats have supported border fencing in the past, so it should be an easy lift.

“The bottom line is everything I just outlined has been offered by each party before,” he said....