Democrats enter this week’s immigration debate with the goal of protecting illegal immigrant Dreamers but with little strategy for how to get there, saying they will take their chances when the fight plays out on the floor of the Senate.

Party leaders don’t have a Democratic plan in hand, nor do they have a sense for what concessions they are willing to make to reach a deal with President Trump, beyond some sort of funding for his border wall.

Republicans, meanwhile, are struggling for unanimity within their own ranks with Mr. Trump’s four-point framework unable to win over all 51 members of his party in the Senate.

“This is a wild card week,” said Sen. James Lankford, an Oklahoma Republican who is co-sponsoring the legislative version of Mr. Trump’s plan but is also working with the bipartisan moderates who have dubbed themselves the Common Sense Caucus and are reaching for their own deal.

The Senate voted 97-1 Monday evening to head off the chance of a filibuster at the start of the immigration debate, setting up action as soon as Tuesday on proposals to deal with Dreamers, security enhancements and bigger changes to the legal immigration system.

Republican leaders are backing a bill that enshrines Mr. Trump’s framework, coupling citizenship rights for up to 1.8 million Dreamers with $25 billion in border wall funding, an end to the catch-and-release policies that free illegal immigrants into the interior of the U.S. and faster deportations for visitors who overstay their permits.