The pair have "chemistry," according to the White House and Indian officials. They've shared impromptu strolls on the National Mall and, on Sunday, a long meander through a New Delhi rose garden.
Indian weavers are hurriedly piecing together saris as gifts for the first lady. The normally congested Indian capital has been rid of its stray monkeys -- at least in the places Obama might visit. And local television networks are running entire pieces speculating the two leaders might take a selfie.
Here's a highlight reel of social media's favorite Obama and Modi broments so far:
In case anyone doubted the deep bond between Obama and Modi (and how could they?), Air Force One's arrival on Sunday put any questions to rest.
Breaking with established protocol -- and surprising television commentators on the dozen or so local news broadcasters -- Modi showed up on the tarmac to be the first to greet Obama in India.
And it wasn't just with a handshake. Modi pulled the travel-weary commander in chief for the "bear hug" seen around the world -- though like most unplanned hugs in front of a billion-viewer television audience, it was a little awkward. The two leaders later followed it up with another warm embrace after a joint press conference, stoking rumors of a "bromance."
When Modi visited Obama at the White House in September, the visit ended with Obama blowing up his set-in-stone schedule and taking the visiting Prime Minister for a stroll at the Martin Luther King memorial on the National Mall.
The strolling continued in New Delhi.
Obama and Modi, punctuating their bilateral talks, sauntered side by side through the formal gardens at Hyderabad House, a stately 1920s home that now acts as a ceremonial venue. Cameras caught them murmuring away to each other, but reporters could not hear what they said.
After the stroll they sat down for a cup of tea. Modi's camp wasted no time posting a Facebook album of highlights.
When Obama and Modi last sat down for dinner, at the White House in September, only one of them actually ate. Modi was midway through a religious fast that allowed only warm water.
This time around, solid foods were consumed by all. A readout of lunch showed vegetarian and nonvegetarian menus, each with more than a dozen dishes.
(Even so, Indian celebrity chef Saransh Goila criticized its lack of variety.)
Dinner was an even bigger affair: Held at the presidential palace, the event featured Indian dancers, a receiving line and, of course, a pair of heartfelt toasts.
India and the U.S. share the "natural sense of kinship" their leaders enjoy, Modi said. Obama went with a passage from Walt Whitman's "Passage to India":
"The earth to be spann'd, connected by network, / The people to become brothers and sisters."
"Here, in our time, these words have come to pass," Obama said.
The leaders poked fun at each other throughout the day. During a press conference, Obama remarked that Modi was greeted like a "Bollywood star" when he visited New York's Madison Square Garden last year.
Obama also said they compared how much sleep they're getting, prompting polite laughter from the press corps.
"And it turns out that Modi is getting even less sleep than me. But of course that's because he's still new. After you've been doing this for about six years, maybe he'll be able to get an extra hour."
Not everyone was impressed with his answer, however.
And this is just the first day of Obama's historic visit. Stay tuned for highlights from Day Two, which has a parade on the itinerary.