The U.S. has said Rusesabagina was unlawfully detained, while Rwanda has bristled at the criticism, saying it would not be intimidated.
The U.S. allocated more than $147 million in foreign assistance to Rwanda in 2021, making it Rwanda's largest bilateral donor.
Blinken met Kagame during a visit to Rwanda in August, where U.S. officials said the case was discussed extensively.
Yet Kigali continued to take a hard line, with Kagame suggesting on the sidelines of the December summit that only an invasion of Rwanda could force Rusesabagina's release.
He will remain in Rwanda for a couple of days before travelling to Doha and then to the United States, U.S. officials said.