The Ukraine conflict has plunged relations between the two nuclear powers to their lowest levels since the Cold War, with Trump repeatedly promising to end the fighting with a "deal".
He told reporters on Thursday he would meet Putin "immediately", and that Ukrainian President Volodymyr Zelensky wanted to negotiate.
"Putin is ready. We are waiting for signals," Kremlin spokesman Dmitry Peskov said of Trump's overtures.
Peskov said that he could not comment further on a possible meeting between the leaders, saying it was "hard to read coffee grounds" to predict the future.
Trump has threatened Russia with tougher economic sanctions if it does not agree to end its nearly three-year offensive.
The Kremlin rejected Trump's claim that the conflict in Ukraine could be ended by lowering the price of oil used to fund Moscow's budget, saying: "This conflict does not depend on oil prices."
Peskov said the conflict was instead based on "threats to Russia's national security", "threats to Russians" living in Ukraine and "the lack of desire and complete refusal of Americans and Europeans to listen to Russia's concerns".
