“The United States notes with great concern the Rwandan senate’s vote,” State Department deputy spokesman Mark Toner told reporters in Washington.
The diplomat did not explicitly threaten that US aid to Rwanda would be cut, but warned ties could be reviewed.
The Rwandan senate’s decision to approve an amendment to the constitution on Monday evening must still go to a referendum.
Kagame is the latest long-serving ruler in Africa to attempt to extend his hold on power. Similar moves have already sparked violence and instability in Burkina Faso, Burundi and Congo Republic. So far there has been no unrest in Rwanda.
Critics accuse Kagame, 57, of trampling on media and political freedoms, charge officials deny. The government says it cannot be blamed if opponents fail to win at the ballot box.
A date for a referendum has yet to be decided.
Kagame has run Rwanda since his rebel army ended the 1994 genocide and ousted Hutu extremists.
He won elections in 2003 and 2010 and, under the current law, is due to step aside in 2017 at the end of his second term.
Earlier this year, more than 60 percent of voters signed a petition calling for constitutional changes to be drafted that would allow Kagame to stand again.