President Joe Biden announced new sanctions on Russia on Tuesday following Russian President Vladimir Putin signaling he would send "military assistance" to the two Russian-backed separatist regions in eastern Ukraine whose independence Moscow has recognized.

"This is the beginning of a Russian invasion of Ukraine, as he indicated and asked permission to be able to do from his Duma," Biden said in remarks from the White House. "So I'm going to begin to impose sanctions in response -- far beyond the steps we and our allies and partners implemented in 2014."

"If Russia goes further with this invasion, we stand prepared to go further as with sanctions," Biden continued.

In his first public remarks since Friday on Ukraine, the president said Putin is "setting up a rationale to take more territory by force" -- and "to go much further."

"Who in the Lord's name does Putin think gives him the right to declare new so-called countries on territory that belonged to his neighbors? This is a flagrant violation of international law and demands a firm response from the international community," he added.

Ahead of an update from President Joe Biden on the situation at Ukraine's border, lawmakers on both sides of the aisle are calling on the president to impose harder sanctions on Russia following criticism that sanctions announced Monday were limited.

Senate Republican Leader Mitch McConnell said in Kentucky this afternoon he hopes Biden will say that the U.S. is "going to impose the toughest possible sanctions."

He also said any path forward should ensure that the Nord Stream 2 pipeline not be allowed to ever proceed.

"So as all of this unfolds let me be perfectly clear: The toughest possible sanctions plus no Nord Stream 2. Not now. Not tomorrow. Not ever," McConnell said.

In an earlier statement, he also called on the U.S. and NATO allies to send support to Ukraine, "including arms," and warned, "The world is watching."

Senate Foreign Relations Committee Chairman Bob Menendez, a Democrat, has also called on the Biden Administration to impose "severe sanctions," telling CNN Tuesday that it's time to "stop equivocating" on whether or not there has been an invasion.

He also said he believes, though he said he couldn't disclose intelligence, that more Russian troops have arrived in Ukraine overnight.

-ABC News' Allison Pecorin

Russia’s Foreign Ministry announced Tuesday it will begin evacuating its personnel from its embassy in Ukraine in "the very nearest time," according to Russian state news agencies.

RIA Novosti reported that the foreign ministry has claimed its staff received threats and that Ukraine has not reacted to them, while Ukrainian officials have maintained that Russia is the aggressor on the ground.

-ABC News' Patrick Reevell

In a press briefing at NATO headquarters in Brussels, NATO Secretary-General Jens Stoltenberg urged Russia to "choose the path of diplomacy," as he said that Russia's aggressive behavior towards Ukraine marks "the most dangerous moment in European security in a generation."

Stoltenberg said that there is evidence that Russia has already entered Ukraine, going as far as saying that Russia has been in Donbas since 2014 in what he considers "covert" operations. He said Russia has now moved from "covert attempts to destabilize Ukraine to overt military action."

Russia has deployed over 150,000 troops, fighter jets and attack helicopters in Ukraine and Belarus and along the Russia-Ukraine border, with troops "in the field and ready to attack," according to Stoltenberg, while NATO allies have deployed more troops in Romania, Estonia, and Lithuania, and more than 120 ships and over 100 jets are on "high alert." The NATO response force is on "high readiness," but has not been deployed.

Stoltenberg said that it is "never too late not to attack," and that options for diplomacy are still available to Russia, even despite Putin's "threatening rhetoric" in his address to the public Monday. "We are ready to talk," said Stoltenberg, as NATO continues to look for a "political path forward."

-ABC News' Christine Theodorou