Senate Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellFormer OMB pick Neera Tanden to serve as senior adviser to Biden Lawmakers reach agreement on bipartisan Jan. 6 commission The Hill's Morning Report - Presented by Facebook - Masks off: CDC greenlights return to normal for vaccinated Americans MORE (R-Ky.) blocked legislation on Thursday that would have reopened most of the federal government impacted by the partial shutdown.

Sen. Tim KaineTimothy (Tim) Michael KaineSenate Democrats ramp up push to limit Biden's war powers Sweeping election reform bill faces Senate buzz saw How leaving Afghanistan cancels our post-9/11 use of force MORE (D-Va.) tried to get consent to take up a House-passed bill that would reopen all agencies except the Department of Homeland Security (DHS), which is at the center of the shutdown fight. But McConnell objected.

The Senate GOP leader didn’t explain his objection from the Senate floor, but he has warned for weeks that he will not bring up a government funding bill unless it’s the product of an agreement between congressional Democratic leadership and President TrumpDonald TrumpVirginia GOP gubernatorial nominee acknowledges Biden was 'legitimately' elected Biden meets with DACA recipients on immigration reform Overnight Health Care: States begin lifting mask mandates after new CDC guidance | Walmart, Trader Joe's will no longer require customers to wear masks | CDC finds Pfizer, Moderna vaccines 94 percent effective in health workers MORE.

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"The solution to this is a negotiation between the one person in the country who can sign something into law, the president of the United States, and our Democratic colleagues," McConnell said Tuesday when he blocked a House bill as well as a piece of legislation to fund DHS through Feb. 8. 

This marks the third time that McConnell has blocked House-passed government funding bills in the past two weeks. Under Senate rules, any senator can ask for consent to vote on or pass a bill, but any senator can object. 

Roughly a quarter of the government has been shut down since Dec. 22, forcing roughly 800,000 federal employees to work without pay or be furloughed. 

Thursday marks the 27th day of the partial shutdown with no obvious path to a quick end to the funding fight.

Talks between Trump and congressional Democratic leadership derailed after a White House meeting last week. The president left after Speaker Nancy PelosiNancy PelosiGohmert says Jan. 6 mob attack on Capitol not an 'armed insurrection' Meghan McCain: Greene 'behaving like an animal' GOP Rep. Turner to lead House push to address military sexual assault MORE (D-Calif.) said she wouldn't negotiate on the border wall even if the president fully reopened the government.

Tensions further escalated this week with Pelosi asking to postpone the Jan. 29 State of the Union speech and the president, in retaliation, canceling a foreign policy trip she had planned.

Vice President Pence and the president's son-in-law and adviser Jared KushnerJared Corey KushnerNew Kushner group aims to promote relations between Arab states, Israel Republicans request documents on Kerry's security clearance process Iran moves closer to a diplomatic breakthrough that may upset Israel MORE met with McConnell shortly after the floor drama on Thursday evening, but the GOP leader indicated they had not reached an agreement during their talks.

"We're still obviously discussing the situation and if I have any news to make, I'll let you know," McConnell told reporters.

The Senate passed a stopgap bill late last year by a voice vote, but it was rejected by the White House because it didn’t include extra border money.

Trump is demanding more than $5 billion for his signature wall. Democratic leadership has pointed to $1.3 billion as their cap and argued that it must go to fencing.

Democrats are trying to build pressure on McConnell to move government funding legislation even without Trump's blessing, something members of GOP leadership say is not going to happen.

But Kaine, whose state has a large population of federal workers, will force the Senate to be in session on Friday and Saturday. The Senate will also be in session Tuesday, Wednesday and Thursday of next week, though senators say votes currently aren't scheduled.

Kaine is planning to ask for consent to pass the House package to reopen the government on each of the days, except Saturday. The move will force McConnell or another GOP senator to be on hand to object.

"The first level of illogic we are in is we are punishing the workers and citizens who need services completely unconnected with the border and immigration reform issue," Kaine said on Thursday, about his request to move the package unrelated to DHS funding.

Several GOP senators have backed either taking up the House bills or passing a continuing resolution to reopen most or all of the government while Democratic leadership and Trump continue to fight over border security.

A bipartisan group of senators, led by Sens. Lindsey GrahamLindsey Olin GrahamGOP governors move to cut unemployment benefits as debate rages over effects Trump critics push new direction for GOP Graham warns about trying to 'drive' Trump from GOP: 'Half the people will leave' MORE (R-S.C.) and Christopher CoonsChris Andrew CoonsBiden to go one-on-one with Manchin US, Iran signal possible breakthroughs in nuke talks How the United States can pass Civics 101 MORE (D-Del.), were drafting a letter to send to Trump to ask him to back a three-week continuing resolution in exchange for the Senate agreeing to take up his border request in the Appropriations Committee as an supplemental appropriations bill.

The idea has received pushback from Republicans and the White House, as well as skepticism from Democrats who are wary of publicly backing the letter without a sense of how much GOP support it has.

But Sen. Susan CollinsSusan Margaret CollinsFormer OMB pick Neera Tanden to serve as senior adviser to Biden The Hill's Morning Report - Presented by Facebook - Masks off: CDC greenlights return to normal for vaccinated Americans Masks shed at White House; McConnell: 'Free at last' MORE (R-Maine), a member of the group, said on Thursday that they had another meeting and were still trying to come up with a proposal along the lines of the one outlined in the letter.

"There are members in good faith who are working on both sides of the aisle, and we continue to have discussions," Collins told reporters. "I've just had a meeting with both some Republican members and some Democratic members to discuss a possible path forward."

Alexander Bolton contributed to this report that was updated at 7:10 p.m.