"There isn't a free moment," says Pusha T on the phone from Virginia. It's 9am on the east coast and the veteran rapper, currently one of the hottest emcees on the planet, has been up bright and early washing clothes. "I'm supposed to be taking a holiday, man, but today I've got to go to New York City for two meetings about my album and another meeting about my clothing line." He sighs. "So much for the vacation."

Pusha T – born Terrence Thornton in the Bronx and raised in Norfolk, Virginia – has been active in hip-hop for the past two decades, but he's never been busier than he is right now. Last October, four years into his solo career, having taken extended leave from the fraternal rap duo Clipse, he released his debut album. My Name is My Name had plenty of wattage behind it – Pusha's label boss, Kanye West, oversaw production and guest stars included Chris Brown and Kendrick Lamar – but nobody expected it to generate much commercial heat. The album sold 74,000 copies in its first week and went to No 4 on the Billboard charts, not bad for a hard-edged rap record largely preoccupied with the travails of being a crack dealer.

Since then, the 36-year-old has been touring intensively in the US. He's joining Kanye West on tour in Australia in early May and this summer he plays a string of festivals around the UK and Europe. Does he enjoy doing big outdoor shows?

"Oh man, I really like it," he says. "You really get to see who your fans really are at those events, because a lot of times there's four, five, six stages going on. You've got competition and those people definitely do not have to be standing in front of your stage."

Controlling festival audiences can be a challenge. "Playing a room, it's a real dictator-type situation – you can really move the crowd the way you want to. In a festival, there's a sea of people and it's harder to lock in on any one group."

Back in the early days of Clipse, which he started in the early 90s with his older brother Gene and built with the help of fellow Virginians the Neptunes, who produced most of their material, Pusha took a looser approach to performing live. "I used to be a guy who'd get off stage and go to the club – drinking, doing a ton of things that I probably shouldn't be doing, which ruins me for the next day." Now he says he's learned how to tour properly. "I've got my travelling, my packing, my after-show activities all down to a science. I used to not work out on tour; now I take a trainer with me. I do things to make sure that I can give the crowd my all, because that's what I'm all about."

It sounds very businesslike. He agrees. "I think about nothing but business the whole time. Even when I go out to my after-parties, people are like, 'You're drinking water?' I'm like, 'Yeah I am.'"

He may have picked up a lesson or two from Kanye West, another rapper with a punishing work schedule. They first crossed paths in 2007 when West asked Clipse to play at his 30th birthday in New York. Three years later, Pusha signed to West's label GOOD Music. He rates his boss as one of three super-producers working in hip-hop, the others being Pharrell Willliams and Timbaland, but Pusha maintains that his own record was superior to Kanye's Yeezus. "I personally don't think anybody made a better album last year," he says breezily.

One of the interesting things about My Name is My Name was how, two decades into his career, Pusha was still rapping about life on the street. Gritty drug-dealing narratives were key to what was so viscerally compelling about Clipse, but doesn't that life feel remote to him now? "No, I don't think that goes anywhere. I still come home to Virginia. I'm still grounded, I still can speak the experience. I still have friends who are in the mix."

He points out that, unlike many rappers who make it big, he didn't move to LA or New York and lock himself away in a mansion. The streets are where he feels most at home. Which makes it even more startling to learn that Pusha T is a big tennis fan. "MAJOR fan," he says. "At the moment, I just watch. One thing I didn't get to do in my supposed vacation was take the lessons I'd booked. They actually expired. I was so upset about that."

If the opportunity arises, he'd love to indulge his passion while he's in the UK this summer. "I caught the US Open a couple of times, but, man, I would love to catch Wimbledon and watch Nadal – he is the king of all kings."

It's not out of the question – Pusha will be in Denmark, playing Roskilde, when Wimbledon is heating up. But it probably won't happen. If his current situation is anything to go by, this sought-after rapper will be simply too pushed for time.

Pusha T appears at the Parklife Weekender in Manchester, 7-8 June, and at the Reading/Leeds festivals, 22-24 August