The Senate could hold the first votes on amendments foras soon as Tuesday afternoon, instead of waiting 30 hours after invoking cloture a day earlier.
Senate Majority Leader Mitch McConnell, R-Kentucky, said Tuesday that there is "no reason" why the Senate can't reach a bipartisan immigration deal this week.
"But to do this, we need to get the debate started, look past making political points, and focus on making law. Making law will take sixty votes in the Senate, a majority in the House, and the president's signature," McConnell said on the Senate floor.
"Yesterday, a number of my colleagues announced a reasonable proposal that I believe is our best chance to make law. It attends to my Democratic colleagues' stated top priority: A compassionate solution for 1.8 million illegal immigrants who were brought to the United States as children," he added. "In exchange, it also delivers on the president's stated conditions. Their solution provides funding to secure the border, reforms extended family chain migration, and recalibrates the visa lottery program."
Most Senate aides believe that at the end of the day, the Senate will consider at most three or four amendments from each side.
By Wednesday, if some of the amendments fail, a bipartisan group could come in and they could introduce some kind of pared-down plan that pairs legal protections for the DACA population with some amount of border security or wall funding.
CBS News' Nancy Cordes contributed to this report.