Trump has also emphasized that Ukraine must decide for itself what to do with its territory.
Europe is concerned that Trump, eager to quickly conclude a peace agreement, will pressure Zelensky to make territorial concessions, a Bloomberg source said.
The British newspaper"Financial Times" and the Kiev correspondent of the news agency "Reuters", citing sources, also report such demands from Putin.
Citing four sources familiar with Trump and Putin's talks in Alaska, the Financial Times reported that Putin offered to freeze the front line in the Zaporizhia and Kherson regions on condition that the Ukrainian army leaves the entire territory of the Donetsk region.
Neither the White House nor Kremlin spokesman Dmitry Peskov responded to the newspaper's request for comment.
"People familiar with Zelensky's thinking" told the newspaper that he would not agree to hand over the Donetsk region, but would"be open to discussing the issue of territories" when he meets with Trump in Washington on Monday.
Zelensky announced after a phone call with Trump on Saturday that he would meet with the US president at the White House in Washington on Monday.
After his talks in Alaska with Putin and subsequent phone calls with Zelensky and European leaders, Trump wrote on the Truth Social platform that everyone agreed that the best way to end the war was to move toward a peace treaty, rather than negotiate a temporary ceasefire.
The Wall Street Journal reported on August 8, citing sources, that Putin was ready to agree to a ceasefire if Ukraine withdrew its troops from the Donetsk region. The front line in other regions would be frozen.