Thursday's votes were on moving forward to start debating each bill, and both needed at least 60 votes to advance, but each failed to garner enough bipartisan support. The GOP proposal failed 51-47 and the Democratic alternative by 52-44 -- with six Republicans crossing the aisle.
The Republican bill mirrored the president's proposal from Saturday -- a three-year extension to Temporary Protected Status recipients and about 700,000 so-called “Dreamers,” undocumented immigrants brought to the U.S. as children and given protected status under DACA, or the Deferred Action for Childhood Arrivals program, in exchange for border wall funding. Democrats and other critics say the president's proposal undercuts some protections for immigrants.
The measure proposed by Democrats didn't include additional funding for a border wall and instead would have funded the government until Feb. 8, giving time for negotiation while getting 800,000 furloughed federal employees back to work and getting paychecks.
Republican Sens. Susan Collins of Maine and Cory Gardner of Colorado said they would vote for both proposals in the procedural vote Thursday. Democratic Sen. Joe Manchin said he would do the same.
Trump announced late Wednesday night that he would hold off on giving a State of the Union address until the government reopens, a decision forced by House Speaker Nancy Pelosi.
The back-and-forth between the president and the House speaker continued Thursday as Pelosi held a news conference urging the Senate to pass a bill ending the shutdown. As she spoke to reporters, the president tweeted out a promise not to "cave" until he gets a border wall.
The president, however, appeared to be responding to Pelosi's comments not about her views on the shutdown but quoting Commerce Secretary Wilbur Ross, who said in an interview he didn't "understand" why federal workers were going to food banks instead of taking out loans. Ross was promptly criticized for downplaying the effects of the partial shutdown on individual workers.
About a quarter of the government remains unfunded including the Department of Homeland Security, prompting a letter from five former secretaries of the department -- including Trump's former chief of staff, John Kelly.
DHS should be no different from the Department of Defense, the former secretaries wrote, which is funded in the shutdown and which is similarly important to defending national security. The secretaries also wrote that the shutdown was tempting talented employees to turn to the private sector -- a problem that grows with each week the shutdown persists.
"High-tech skills – such as information technology and cybersecurity - and law enforcement and security experience are in great demand in the private sector and the Department is facing a real crisis in retaining this workforce week after week," wrote former DHS Secretaries Tom Ridge, Michael Chertoff, Janet Napolitano, Jeh Johnson and Kelly.
The IRS, also not funded during the shutdown, is also facing difficulties among the legions of employees who are working without pay -- now 46,000 of them after a recent recall in attempt to mitigate the impact of the shutdown, especially ahead of pressure to process tax returns.
IRS employees have begun claiming they can't fulfill the request to come back to work for the upcoming tax season because of financial hardship, a major federal employees' union said Wednesday. The move could slow down the agency's ability to issue tax returns and also ratchet up pressure on Trump and Congress to push through budget talks.
On Friday, many federal workers will miss their second paycheck because of the shutdown. More federal workers will miss a second paycheck on Monday, and by Tuesday, all 800,000 federal workers impacted by the partial shutdown will have missed a second straight paycheck. About a half-a-million of these federal employees are working despite missing pay.
The current shutdown is the longest in U.S. history, surpassing a 21-day shutdown that ended in 1996.
ABC News' Alexander Mallin, Mariam Khan, Trish Turner, John Parkinson, Anne Flaherty, Stephanie Ebbs, Quinn Owen and Luke Barr contributed to this report.