The short answer is that yeah, there's basis for the idea itself (as in someone other than Sienkiewicz came up with it), but only slim basis to say that Nero started the fires—and that the idea that he did it emerged no later than about half a century after Nero's death.

The long answer is that the historian Suetonius (writing during the reign of Hadrian) claimed that Nero's agents set fire to the city. His accusation is grand, and includes a depiction of the intentional demolition of buildings using siege equipment. Suetonius relays this story in his Lives of the Caesars, translations of which are available online. The notion that people working for the emperor actively lit fire to things is subsequently repeated by the senator Cassius Dio in his History of Rome, writing around a century later. Book 62 of Dio's work, which covers the reign of Nero, survives in bits and pieces thanks to a monk writing in the 11th century. What we know about the event is a bit filtered, then, and might all ultimately go back to one original source (being Suetonius).

Suetonius is interesting, because he wasn't just a historian—he was Hadrian's personal secretary, and he was involved in some funny business involving Hadrian's wife, probably around 122 CE, just a few years after Hadrian's ascension. That date is guessed at (by Birley, mentioned in a bit) based on the the concurrent dismissal of a certain prefect of the guard, also involved in that same funny business. If so, this means Suetonius spent the next sixteen years until Hadrian's death not being Hadrian's secretary. He was, in the meantime, writing his Lives of the Caesars, which quite understandably did not document the Caesar then ruling Rome—crossing Hadrian was dangerous, even for artists and scholars, and Suetonius was already in disfavor. Anthony Birley suggests in Hadrian: The Restless Emperor that Suetonius intended his history of Domitian (the last of his twelve, perhaps written during the waning years of Hadrian's reign) to be an indirect critique of Hadrian's more oppressive (read: anti-Senate) activities as emperor. It's possible his history of Nero was similarly intended—there are more than a few parallels between the two bearded, artistic Philhellenes (one historian calls Hadrian "a good Nero"), so presenting Nero in a critical light could have been a way for Suetonius to imply further parallels.

I have a gap in what I know, and that is Tacitus' account of things—and he apparently gives Nero an alibi putting him outside of the city when the fire happened. Regardless, Nero as the mastermind behind the fire is unlikely. The simplest analysis is that Rome burned a lot, it burned in a big way when Nero was emperor, and Nero wasn't popular with pro-Senate types who wrote history.