Russian forces are battling to eject Ukrainian troops from the border region.
On Wednesday, Russian troops raised their flags over central Sudzha as Ukrainian forces hurriedly retreated back toward the shared border.
Russia's battlefield successes in Kursk come as the U.S. pushes both Moscow and Kyiv to return to peace negotiations. This week, Ukraine and the U.S. agreed to a potential 30-day ceasefire, with American representatives also putting the proposal to a non-committal Kremlin.
Russian officials have indicated that they will not engage in peace negotiations while any of Kursk remains under Ukrainian control. Kyiv had hoped to use its occupation of the territory as leverage in talks, though its footprint there is now rapidly shrinking.
On Wednesday, Putin said he will give "special thought in the future to creating a security zone along the state border" to prevent repeat Ukrainian incursions. Prisoners taken on Russian territory would be treated "as terrorists," Putin said, adding that "foreign mercenaries" are not protected under the Geneva Conventions.
President Donald Trump's Middle East envoy Steve Witkoff is expected in Moscow this week as the administration pushes for a ceasefire and broader peace deal. The ball is now "truly in their court," Secretary of State Marco Rubio said of Russia following the U.S.-Ukrainian agreement to a 30-day ceasefire proposal.
The Kremlin was non-committal. Officials were "scrutinizing" the publicly released statements, Kremlin spokesperson Dmitry Peskov said on Wednesday. Russia, he added, "doesn't want to get ahead of itself" on the potential ceasefire.
On Thursday, Peskov confirmed that American negotiators are traveling to Moscow. "Contacts are planned," Peskov told a press briefing, adding of the potential outcomes, "We will not prejudge, we will tell you later." Peskov did not say whether Witkoff would meet with Putin.
Trump's push for peace -- which has been twinned with fierce public criticism of Ukraine and President Volodymyr Zelenskyy -- has been welcomed by America's allies, though leaders have been perturbed by the president's apparent alignment with Russia's false narratives about the conflict.
Rubio will meet with G7 foreign ministers in Quebec, Canada, on Thursday. His presence at the meeting will also be overshadowed by Trump's spiralling trade war with America's northern neighbor, plus the president's repeated suggestion that Canada be absorbed by the U.S. and become its 51st state.
The G7 event "is not a meeting about how we're going to take over Canada," Rubio said Wednesday, as quoted by the Associated Press.
Canadian Foreign Minister Melanie Joly, though, said that "in every single meeting, I will raise the issue of tariffs to coordinate a response with the Europeans and to put pressure on the Americans."
"The only constant in this unjustifiable trade war seems to be President Trump's talk of annexing our country through economic coercion," Joly said. "Yesterday, he called our border a fictional line and repeated his disrespectful 51st state rhetoric."
ABC News' Tanya Stukalova, Patrick Reevell and Will Gretsky contributed to this report.
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