"It is in the fundamental interests of the two countries and peoples for China and India to correctly grasp the trend of history and the direction of development of their relations," Xi said, according to Chinese state broadcaster CCTV.
In response, Modi told Xi that maintaining peace and stability on their frontier should be a priority, and that mutual trust, respect and sensitivity should be the basis of the relationship.
"We welcome the agreement on the issues that had come up over the last four years," Modi told Xi in comments that were aired on India's state broadcaster Doordarshan.
BRICS - an idea thought up inside Goldman Sachs two decades ago to describe the growing economic clout of China and other major emerging markets - is now a group that accounts for 45% of the world's population and 35% of the global economy.
The BRICS summit clashes with the International Monetary Fund and World Bank annual meetings in Washington.
Former Goldman economist Jim O'Neill, who coined the BRIC term in 2001, told Reuters that he had little optimism for the BRICS club as long as China and India remained so divided.
"It seems to me basically to be a symbolic annual gathering where important emerging countries, particularly noisy ones like Russia, but also China, can basically get together and highlight how good it is to be part of something that doesn't involve the US and that global governance isn't good enough," O'Neill told Reuters.
UKRAINE WAR
Putin, who dismisses Western claims that he is a war criminal for Russia's actions in Ukraine, hosted more than 20 leaders at the summit in the city of Kazan on the banks of the Volga, including Tayyip Erdogan, who leads NATO member Turkey, and Iranian President Masoud Pezeshkian.
Modi told Putin in public that he wanted peace in Ukraine. Xi discussed the war in the Ukraine behind closed doors with the Kremlin chief, as did United Arab Emirates President Sheikh Mohammed bin Zayed Al Nahyan who has sought to mediate.
The 43-page final communique from the summit ranged from geopolitics and narcotics to artificial intelligence and even the preservation of big cats, but lacked detail on some major issues. It mentioned Ukraine just once.
"We note with appreciation relevant proposals of mediation and good offices, aimed at a peaceful resolution of the conflict through dialogue and diplomacy," the Kazan declaration said.
The toughest language was reserved for the Middle East, calling for a ceasefire in the Gaza Strip and the West Bank, and denounced Israeli "attacks against humanitarian operations, facilities, personnel and distribution points."
On alternative payment systems, there was little detail, though BRICS leaders expressed an interest in developing them and ordered central bankers to report back under the next presidency.
"The trend for the BRICS' leading role in the global economy will only strengthen," Putin said, citing population growth, urbanization, capital accumulation, and productivity growth as key factors.
China and India buy about 90% of Russia's oil - Moscow's biggest foreign currency earner. Russia is the world's second largest oil exporter.
The leaders said they would seek to further develop BRICS' institutional development, but gave little clarity on whether or not BRICS would be expanded.
Putin said that more than 30 states had expressed interest in joining the group but that it was important to strike a balance in any expansion.
"I will take the BRICS group seriously when I see signs that the two countries that really matter, China and India, actually really trying to agree on things, rather than effectively trying to confront each other all the time," O'Neill said.