Those Russian "peace" demands include Ukraine recognizing Russia's conquest of a fifth of its territory.
Kyiv, armed and supported by the United States and its NATO allies, has vowed to recover all occupied territory and to drive out all Russian soldiers.
Over the past 24 hours, Ukrainian forces have repelled Russian attacks in the areas of two settlements in Luhansk province and six in Donetsk, the General Staff of Ukraine's Armed Forces said on Tuesday.
Britain's defense ministry said in its latest update of the military situation in Ukraine that fighting was particularly intense around the strategic eastern city of Bakhmut in Donetsk province and Svatove, further north in Luhansk province. Donetsk and Luhansk, which make up the industrial Donbas, are both claimed, along with two southern Ukrainian regions, by Russia.
"Russia continues to initiate frequent small-scale assaults in these areas (of Bakhmut and Svatove), although little territory has changed hands," the British ministry tweeted.
"Our proposals for the demilitarization and denazification of the territories controlled by the regime, the elimination of threats to Russia's security emanating from there, including our new lands, are well known to the enemy," TASS news agency quoted Russian Foreign Minister Sergei Lavrov as saying late on Monday.
Reuters footage showed fires burning in a large residential building in Bakhmut, while debris littered the streets and most buildings had had their windows blown out.
In his nightly video message on Monday, Ukraine's President Volodymyr Zelenskiy called the situation along the frontline in Donbas "difficult and painful".
After suffering a series of defeats in its "special military operation," Russia is now seeking a battlefield victory by capturing Bakhmut, an industrial city with a pre-war population of 70,000, now reduced to about 10,000 mostly elderly residents.
Gaining control of the city could give Russia a stepping stone to advance on two bigger cities, Kramatorsk and Sloviansk.
It also reported more Russian shelling of Kherson city, in the Zaporizhzhia region and of settlements in the Kharkiv region of northeast Ukraine near the border with Russia.
Oleh Zhdanov, a military analyst based in Kyiv, also cited heavy fighting around elevated areas near Kreminna in Luhansk region as well as around Bakhmut and Avdiivka in Donetsk.
"The arc of fire in Donetsk region continues to burn," Zhdanov said in a social media video post.
Putin repeated on Sunday that he was open to negotiations but only on Moscow's terms, a stance Kyiv and the West have rejected. Kyiv and Western countries say Russia is engaged in a brutal, imperialist land grab.
The United States and its allies have imposed sweeping sanctions on Russia and sent billions of dollars in assistance to the Ukrainian government.
Just last week, as Zelenskiy visited Washington, the United States announced another $1.85 billion in military assistance for Ukraine, including a transfer of the Patriot Air Defense System, angering Moscow.
"It is no secret to anyone that the strategic goal of the United States and its NATO allies is to defeat Russia on the battlefield as a mechanism for significantly weakening or even destroying our country," Lavrov told TASS on Monday.
Both President Joe Biden and Vlodomyr Zelenskyy have refuted Moscow's claims, saying they are focused on restoring Ukraine's sovereignty and borders, not topple Russia.