In August, the US first lady wrote a letter to Mr Putin and had husband Donald Trump hand deliver it during his meeting with the Russian president in Alaska.
Russia's invasion of Ukraine has resulted in Russia taking Ukrainian children out of their country so that they can be raised as Russian, it is alleged.
The first lady, speaking to reporters at the White House on Friday, said Mr Putin responded to her letter and after that, they had established an "open channel of communication" regarding the welfare of those children.
"We have agreed to co-operate with each other for the benefit of all people involved in this war," she said.
The Associated Press documented the taking of Ukrainian children in 2022, after which the International Criminal Court said it had issued an arrest warrant for Mr Putin for war crimes, accusing him of personal responsibility for the abductions of children from Ukraine.