Israeli police said the man was suspected of stealing work from several unnamed international stars.
A private investigator connected to the case confirmed Madonna was among them.
The pop star rush-released a number of new songs last month after demos and unfinished mixes from her new album Rebel Heart made their way online.
Madonna called the leak 'a form of terrorism.'
Her manager Guy Oseary said the 'devastating' action had prompted the surprise release of six songs from Rebel Heart in December, months earlier than planned.
Asher Wizman, the owner of a private investigation firm in Israel, told the Reuters news agency he was contacted by Madonna's team several weeks ago after rumours of an Israeli connection to the leak.
'Our investigator found her computers, at home and at a studio, were broken into from a computer in Israel,' he told Reuters.
'We tracked down the computer, and the man behind it. After gathering enough evidence, we turned to the police and he was arrested today.'
Police did not immediately name the 39-year-old man, although local reports identified him as a former contestant on one of Israel's TV talent shows.
'He is suspected of computer hacking, copyright violation and fraudulent receipt of goods,' a police spokesman said.
'During the investigation it appeared the suspect had broken into the computers of a number of international artists, stole unreleased demos and final tracks and sold them over the internet.'
The arrest in Israel comes a day after Bjork was forced to release her new album Vulnicura two months ahead of schedule after it leaked over the weekend.
From BBC News