Sumy borders Russia's Kursk region, where a surprise Ukrainian incursion in 2024 captured a pocket of land in the first occupation of Russian territory since World War II.
Creating a buffer zone could help Russia prevent further Ukrainian cross-border attacks there.
Meanwhile, a Russian bombing campaign that had escalated in recent days slowed overnight as far fewer Russian drones targeted Ukrainian towns and cities.
Moscow's invasion has shown no signs of stopping despite months of intense US-led efforts to secure a ceasefire and get traction for peace talks.
Since Russian and Ukrainian delegations met in Turkey earlier in May for their first direct talks in three years, a large prisoner exchange has been the only tangible outcome, but negotiations have brought no significant breakthrough.
Between Friday and Sunday, Russia launched about 900 drones at Ukraine, officials said, amid a spate of large-scale bombardments.
On Sunday night, Russia launched its biggest drone attack of the three-year war against Ukraine, firing 355 drones.
From Monday to Tuesday, Russia fired 60 drones at Ukraine, the Ukrainian air force said.
Russia's defence ministry claimed its air defences downed 99 Ukrainian drones overnight over seven Russian regions.
The weekend surge in Russia's bombardments of Ukraine drew a rebuke from US President Donald Trump, who said Putin had gone "crazy".
That comment prompted a sharp Kremlin reaction on Monday, with spokesman Dmitry Peskov criticising "emotional reactions" to events.
Peskov adopted a milder tone Tuesday, hailing US peace efforts and saying that "the Americans and President Trump have taken a quite balanced approach".
In Sumy, Russian forces are trying to advance deeper after capturing villages, Oleh Hryhorov, head of the Sumy regional military administration, said in a statement.
Ukrainian forces were endeavouring to hold the line, he said.
Residents of the captured villages were evacuated earlier, and there is no immediate threat to civilians, Hryhorov said.
Putin visited the Kursk region last week for the first time since Moscow claimed in April that it drove Ukrainian forces out of the area where they captured land last August.
Kyiv officials have denied the claim.
The long border remained vulnerable to Ukrainian incursions, Putin said.
He said he told the Russian military to create a "security buffer zone" along the border but provided no public details of where the proposed zone would be or how far it would stretch.
Putin said a year ago that a Russian offensive at the time aimed to create a buffer zone in Ukraine's northeastern Kharkiv region.
That could have helped protect Russia's Belgorod border region, where frequent Ukrainian attacks have embarrassed the Kremlin.
