English Premiership club Saracens said they were aware of an incident involving their player and would be dealing with the matter internally.
The Mallorca-based Ultima Hora newspaper said Vunipola, 31, was tasered twice by police after arriving at a bar in the island's capital, Palma.
The report said he had at one point removed his shirt and confronted passers-by, adding that it took eight police officers to subdue and handcuff him.
He was then taken to a hospital where he had to be sedated, it added.
The newspaper's website carried a video showing a handcuffed Vunipola, as well as other men, being led from a police van into a courtroom.
A Saracens statement on social media said: "Saracens is aware of an incident involving Billy Vunipola in Majorca.
"We will of course deal with this incident internally, and will not make any further comment until then."
Vunipola is expected to leave Saracens at the end of this season and has previously been linked with a move to France, although nothing has been confirmed.
He has won 75 England caps, making his Test debut against Argentina in 2013. He is not currently part of the England squad.
Vunipola made an appearance as a second-half substitute for Saracens in their 15-12 victory over Bath on Friday.
The club are not scheduled to play again until May 11.
Saracens are second in the English Premiership and are chasing a sixth league title in last 10 seasons with just two rounds of regular season games left before the play-offs.
