Trump announces new impeachment legal team

Both Senate leaders focused their floor remarks Tuesday on the Senate power-sharing agreement -- which was agreed to Monday night when Minority Leader McConnell, feeling assured that Democrats do not have the votes to overturn the filibuster rule, agreed to move forward without language explicitly guaranteeing that the rule will stay in place.

According to Majority Leader Chuck Schumer, McConnell "relented".

"I'm glad we're finally able to get the Senate up and running -- my only regret is that it took so long," Schumer said.

COVID-19 relief will now come into focus as a key priority for Democrats -- with former President Donald Trump's impeachment trial looming.

Schumer left open the possibility of using other options to act on COVID-19 priorities if Republicans will not cooperate, saying the Senate will move "without them if we must."

The whole of McConnell's speech was focused on power-sharing and the importance of the legislative filibuster -- which Republicans could use to obstruct Biden's agenda and Democrats could vote to end.

McConnell is banking on Democratic Sens. Kyrsten Sinema of Arizona and Joe Manchin of West Virgina's assurances that they will not change their minds on keeping the filibuster rule intact.

"If the Democratic majority were to attack the filibuster they would guarantee themselves immediate chaos," McConnell said.

The stalemate reflected the logistical challenges of a 50-50 Senate in which Democrats carry power since Vice President Kamala Harris, as president of the Senate, has the power to cast tie-breaking votes.

-ABC News Allison Pecorin

In the Senate chamber at about 2:15 p.m., a rarely used motion that calls all senators to their seats in the chamber --  "a live quorum call" -- is set to take place to ensure they all can stand together at once, raise their hand and take their oath as jurors in former President Donald Trump's second impeachment trial.

Around 2:30 p.m, Former President Pro Tempore Chuck Grassley, R-Iowa, will swear in his successor, Sen. Patrick Leahy, D-Vt., who will preside over the first-ever impeachment trial of a former president.

Leahy -- instead of Supreme Court Chief Justice John Roberts -- will then swear in all senators.

Senate Majority Leader Chuck Schumer told MSNBC's Rachel Maddow in an interview Monday night that Roberts wouldn't be presiding over the trial because Trump is no longer the sitting president and confirmed that Leahy, although the presiding officer, will also be able to vote.

"The Constitution says the chief justice presides for a sitting president. So that is not going to be -- so it was up to John Roberts whether he wanted to preside with a president who’s no longer sitting -- Trump -- and he doesn’t want to do it," Schumer said. "So traditionally what has happened is then the next in line is the Senate pro tem -- that’s the most senior senator on the majority side, and that’s Sen. Leahy, who’s a very experienced man and a very fair man."

A spokeswoman for Roberts Monday morning said the chief still officially has no comment.

-ABC News' Trish Turner and Devin Dwyer

National Economic Council Director Brian Deese told ABC News Senior White House Correspondent Mary Bruce in the White House driveway Monday morning that lawmakers wanting a lower-priced COVID-19 proposal from Biden haven't said what that looks like but suggested there may be wiggle room on the amount of the direct payments to Americans.

In an appearance moments before on CNBC, Deese said a few times that the Biden administration is "looking for people's input," referring to negotiations with lawmakers over the $1.9 trillion COVID-19 relief deal.

After a phone call Sunday with 16 bipartisan senators and some representatives from the House Problem Solvers' Caucus, another senior administration official said "the feedback was constructive across the board" -- but on Tuesday, Deese seemed to be soliciting counter-proposals from Republicans to make the package more palatable to them.

"We welcome the focus on targeting, including from Republican members and those provisions should have broad bipartisan support," Deese said. "Certainly if there are ways to make that provision and other provisions more effective, that's something we're open to, that we'll have conversations about."

While Deese emphasized that the Biden administration is looking for ideas on how to tweak the package, he continued to emphasize the urgency to act soon. Biden said Monday he expects the negotiations could continue another "couple of weeks."

The White House on Monday morning updated Biden’s schedule to include 4:45 p.m. remarks “on the fight to contain the COVID-19 pandemic.”

-ABC News' Senior White House Correspondent Mary Bruce and Sarah Kolinovsky

Republicans are increasingly supportive of the belief that this trial is unconstitutional, and at least one will challenge the proceedings Tuesday.

Just before the Senate breaks for lunch around noon, Sen. Rand Paul, R-Ky., will have about 15 minutes to make his argument as to why he thinks the impeachment trial is unconstitutional. He could raise a constitutional point of order -- which should spark a vote -- forcing senators to go on the record which could signal for the first time where most Republicans stand on convicting Trump.

On Tuesday afternoon, constitutional law professor Jonathan Turley -- who was called as a witness by Republicans in Trump's first impeachment trial and argued against his impeachment then -- has been invited to speak at the weekly closed-door GOP lunch. While it isn't clear if Senate Minority Leader Mitch McConnell was involved in inviting Turley, he likely wouldn't be at the Capitol Hill lunch if McConnell disapproved. 

Turley recently wrote that "retroactive impeachment" -- or impeachment or a former president -- is not constitutional. However, the professor himself noted that he has changed his views over time and previously thought it was constitutional.

The absence of Supreme Court Chief Justice John Roberts is another a key factor in many Republicans' case against the legitimacy of the proceedings as Sen. Patrick Leahy, D-Vt., is set to preside over the swearing-in of senators as jurors Tuesday afternoon.

-ABC News' Trish Turner and Devin Dwyer