India’s new prime minister, Narendra Modi, said in a television interview broadcast ahead of his trip to the U.S. this week that he expects relations between New Delhi and Washington to improve and become closer.

“India and the United States are bound together, by history and culture,” Mr. Modi told CNN. “These ties will deepen further.”

In the interview, recorded on Sept. 13 and aired Sunday, Mr. Modi, who was denied a U.S. visa after religious rioting in the state he governed in 2002, acknowledged that there have been “ups and downs” in relations.

But he said that “the mood of both Delhi and Washington is in harmony,” adding: “Both sides have played a role in this.”

Mr. Modi, who came to power in May, has been engaged in a flurry of diplomatic activity – meeting leaders of neighboring countries, traveling to Japan and hosting Chinese president Xi Jinping.

Last week’s visit to India by Mr. Xi – during which China pledged $20 billion in investments in India – was largely overshadowed by tensions along the two countries’ disputed Himalayan border.

As the two leaders were meeting in Delhi, hundreds of Chinese and Indian security forces were facing off in the Ladakh region after Indian authorities say Chinese troops tried to build a road on territory India views as its own.

Wariness of China, which has grown stronger and more assertive as its economy has boomed, is an important motivation for India’s interest in closer relations with the U.S. and its ally, Japan.

For its part, Washington is keen to see a democratic India that is economically and militarily more powerful so it can serve as a more effective counterweight to a rising China.

In the CNN interview, Mr. Modi expressed optimism that India’s economy would rebound after a period of slow growth. “We have fallen from where we were,” he said. “Now we have a chance to rise again.”

Mr. Modi said he believes Indians have “unlimited talent” and enormous entrepreneurial energy. “I have a clear roadmap to channel it” to help boost economic growth, he said.