Trump announces new impeachment legal team

Sen. Rand Paul, R-Ky., on Tuesday will make a point of order to dismiss former President Donald Trump's impeachment trial in the Senate, forcing Republicans to go on the record, indicating how they might vote in a trial.

Asked by Capitol Hill reporters Monday if he would force a vote on the trial's constitutionality, Paul said, "Yes."

Then asked if he has an idea of how many of his Republican colleagues are going to support him, Paul said, "I think it'll be enough to show that you know more than a third of the Senate thinks that the whole proceeding is unconstitutional -- which will show that ultimately they don't have the votes to do an impeachment."

-ABC News' Trish Turner and Allison Pecorin

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