Senate Majority Leader Mitch McConnellAddison (Mitch) Mitchell McConnellBiden, Eastland and rejecting the cult of civility California governor predicts 'xenophobic' GOP will likely be third party in 15 years This week: Congress set for clash on Trump's border request 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 KaineSchumer: Trump must get congressional approval before any military action against Iran Trump faces skepticism about Iran war authority from both parties Overnight Defense: Shanahan exit shocks Washington | Pentagon left rudderless | Lawmakers want answers on Mideast troop deployment | Senate could vote on Saudi arms deal this week | Pompeo says Trump doesn't want war with Iran 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 John TrumpConway defends herself against Hatch Act allegations amid threat of subpoena How to defuse Gulf tensions and avoid war with Iran Trump says 'stubborn child' Fed 'blew it' by not cutting rates 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 PelosiThis week: Congress set for clash on Trump's border request Judd Gregg: An Irish friend and wisdom Juan Williams: Warren on the rise 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 KushnerThe Hill's Morning Report - Crunch time arrives for 2020 Dems with debates on deck Dershowitz: With 'Mideast Marshall plan,' Abbas can help — or hurt — Palestinians Palestinian leaders reject Kushner's economic plan for region 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 senators divided over approach to election security GOP lawmakers want Mulvaney sidelined in budget talks Trump urged to quickly fill Pentagon post amid Iran tensions MORE (R-S.C.) and Christopher CoonsChristopher (Chris) Andrew CoonsDemocrats want White House hopefuls to cool it on Biden attacks Senators revive effort to create McCain human rights commission Senate Dem to reintroduce bill with new name after 'My Little Pony' confusion 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 CollinsMaine House speaker announces challenge to Collins Senate seat GOP senators divided over approach to election security GOP lawmakers want Mulvaney sidelined in budget talks 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.