For all the talk about the mutual admiration between Donald Trump and Vladimir Putin, it may not be Russia that the U.S. president sees as America's critical new ally.

In his first few days in office, Trump scheduled a phone call with Narendra Modi, making the prime minister of India one of the first leaders he spoke to following his inauguration.

In its official statement Tuesday evening, the White House said that Trump and Modi "discussed opportunities to strengthen the partnership between the United States and India in broad areas such as the economy and defense."

Sources close to the prime minster said the conversation was focused on defense. The White House did not respond to a CNBC request for further comment.

A tightening of relations with India is something that was already accelerating under Barack Obama, whose administration saw the world's biggest democracy as a counterbalance to China's rising power. Trump may take the relationship further.