For eight months now, since the end of a relationship, The Black Keys drummer Pat Carney has been living in New York's Lower East Side. He likes being able to buy freshly squeezed orange juice and have conversations with people he's never met before. "I'm much happier here, for the most part," he says.

This morning, though, he's dwelling on Akron, Ohio, the town he left behind. The local basketball team that he supports, the Cleveland Cavaliers, is about to lose its best player, LeBron James. It's now been six decades since the city won a major championship. "It's the curse of Cleveland," he says. "LeBron's gone. He's not going to stick around. I wouldn't." He pauses. "I didn't."

Singer and guitarist Dan Auerbach still lives in Akron with his wife and daughter, down the street from his parents. His grandma's house, where his uncles taught him to play bluegrass, is a few blocks away. He's only in New York for a few days, to open for Pearl Jam at Madison Square Garden. He looks tired, but that's nothing new – he had bags under his eyes on the sleeve of their debut album, eight years ago.

"It's heartbreaking. Our teams get so close and then... " Auerbach tails off. Carney picks up: "It's the same thing with us. We get in Rolling Stone, on a major label and it's like 'they could have been the biggest band'." That moment has arrived: their new, critically lauded album, Brothers, entered the US album chart at No 3.

"It's the Akron curse: you've gotta get out to win," Auerbach offers. Will he leave too? "Thought about it. Thinking about it. The only thing keeping me there is my family." At gigs, he tells the crowd "we're the Black Keys, from Akron, Ohio."

Despite the jokes they make at their hometown's expense, loyalty to it is a crucial part of their identity. Chrissie Hynde left in a hurry and later wrote "My City Was Gone" about Akron's decline. Devo moved to Los Angeles. The Black Keys stayed.

They wanted to record Brothers in Memphis, at Sam Phillips Studios, but ended up in Muscle Shoals, Alabama. In the late 1960s, Rick Hall established this tiny city as a place where great soul music was made, attracting Aretha Franklin, Wilson Pickett and Otis Redding to sing with the superb house bands he assembled.

Muscle Shoals Sound, founded by those session men, turned out to be a disappointment. "It didn't have any of the equipment in it that it used to have," Auerbach says. "All of the sound treatment had been ripped out. It used to be all carpeted floors and burlap walls, alcoves for the amplifiers.

Carney's version is, typically, less diplomatic: "What happened is we hired this engineer who knows his stuff but is nuts and has a bad attitude. So because he had a problem with the guy at Sam Phillips he told us it was broken, when it wasn't. Every break we get, he's trying to buy the studio, or steal shit off the guy who owns it. He was trying to tell us how to make a record." Carney shakes his head indignantly.

Collaborating does not come naturally to The Black Keys. For their first album, The Big Come Up, they did everything themselves, down to the cover art. The label hired a producer for their second, Thickfreakness, but they thought it sounded like soft rock, so they ditched the masters and re-recorded it in a day. Their third, Rubber Factory, was made in an abandoned tyre plant in Akron.

All three records have a raw, sparse sound, turned up as far as distortion will allow. Auerbach's voice is a phenomenal instrument, rough and sweet. It's not hard to hear why Robert Plant is a fan. His guitar playing is elemental, Skip James amplified. Carney's approach to drumming is to remember that "the groove is king" and hit as hard as he can. It still rankles that, for years, every review compared them to The White Stripes.

Even now, the one subject guaranteed to rile Auerbach is his perceived debt to the blues. They chose Fat Possum Records because of its roster, including R L Burnside, Junior Kimbrough and T-Model Ford. Kimbrough is a particularly important influence – Auerbach made three unsuccessful pilgrimages to Mississippi to play with him and released an EP of covers after he died.

"When I listen to our records, I don't hear blues music," he says. "It's ridiculous to say we play blues music." I mention that a new track, "Next Girl", sounds like Muddy Waters to me. "It's weird because we ripped off a hip-hop song, directly. We do that and you think it's a blues song." He shrugs, a little annoyed. Although many of their older fans have surely come by way of a Free or Led Zeppelin reference, it's a lineage The Black Keys want no part of.

The comparison is much less valid than it used to be. On Attack & Release, they teamed up with producer Brian Burton, better known as Danger Mouse. It proved to be an inspired combination – three hip-hop fans making classic rock and soul together. The songs were intended for Ike Turner, but when he died, Auerbach recorded the vocals himself.

"We've been asked to do that a lot of times," Carney says, dismissively. "Helping someone who's made years of bad music make good music is bullshit." A project with ZZ Top fell through. Rod Stewart told Rolling Stone he wanted to work with them, but nothing came of it. "It was just talk. He's been wearing no sleeves, so much hairspray and getting so many blow jobs. Nothing that comes out of that guy's mouth is believable."

Their superb cover of Jerry Butler's "Never Give You Up" is the latest reminder that The Black Keys are, first and foremost, a soul band. "It sounds like a soul record not because we cut it at Muscle Shoals but because we were listening to lots of soul, lots of hip-hop," Auerbach suggests. Last year, on the Blakroc album, they realised a long held ambition, backing rappers Mos Def, RZA, Q-Tip and Raekwon.

They also took a break to work on side-projects. Auerbach's solo debut, Keep it Hid, further opened his mind to the possibilities of collaboration. Carney's band, Drummer, turned out to be a "nightmare" of egos and festering grievances. "It made me hate music for a minute," he says. He has no interest in diving into the New York scene.

For the first time ever, they've taken two other musicians on tour with them, to add bass and Farfisa organ. Live at the Housing Works Cafe, at an AIDS benefit show, they play a ferocious, funky set. Hearing the off kilter keyboards and deep bassline of "Too Afraid to Love You", Auerbach's stated ambition to be sampled by the Wu Tang Clan makes perfect sense.

"We could do the new songs as a two-piece but it's fun to play them like they are on the record," Auerbach says. "These guys understand where we're coming from, they love hip-hop, they know that the groove is king. It's hard to teach people that."