Kagame, in power since 2000, won widespread praise for rebuilding the landlocked Central African country since a 1994 genocide killed about 800,000 people, most of them ethnic Tutsis and moderate Hutus.
While praising Rwanda's economic and social development since then, rights groups say the government severely restricts freedom of expression and does not tolerate dissent, charges the government has denied.
Kagame, 58, 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 Burundi, Burkina Faso and Congo Republic.
So far there has been no political unrest in Rwanda.
Senate head Bernard Makuza said he expected the changes to be approved in a referendum, adding Kagame, whose current term ends in 2017, could run for another seven-year term and then two five-year terms, possibly keeping him in power until 2034.
"But it all depends on his will," he told reporters after the Senate's unanimous vote.
Kagame has not said whether he would run again, but has said he was open to persuasion.
In Washington, the U.S. State Department expressed "great concern" with the Senate's decision and said Kagame should step down at the end of his term in 2017.
"We expect President Kagame to follow through on the commitments he has made previously to foster a new generation of leaders in Rwanda and to step down at the end of his current term in 2017," State Department spokesman Mark Toner said.
Asked later if the United States, a major donor to Rwanda, was considering suspending or reducing U.S. assistance to Kigali, Toner replied: "Not at this time."
He said, however, that if Kagame does decide to stay beyond his current term, it could impact U.S.-Rwanda relations.
The draft amendments to the constitution have already been approved by Rwanda's lower house of parliament.
The Democratic Green Party, Rwanda's only genuine opposition party, which is tiny and has no seats in parliament, tried to block the amendments through legal action, but the Supreme Court rejected its bid, saying it was up to citizens to decide.
Frank Habineza, Democratic Green Party's leader, told Reuters by telephone, that the Senate approval would not deter his party from opposing the amendments. "We will continue with our no-change campaign."
Critics accuse Kagame, 57, of trampling on media and political freedoms, a charge officials deny. The government says it cannot be blamed if opponents fail to win at the ballot box.
Parliament, dominated by Kagame's allies and supporters, debated the issue after a petition calling for changes was signed by 3.7 million supporters of the rebel-turned-president.
A date for a referendum has yet to be decided.
In Burundi, President Pierre Nkurunziza's decision in April to run for a third term triggered protests, killings and a failed coup. Opponents said the move violated a deal that ended a civil war there.
In Congo Republic, voters backed a change to allow President Denis Sassou Nguesso to run for a third consecutive term. The opposition had called for a boycott of that vote and numerous protests have ended in violence.
