Grammy-winning jazz bassist Marcus Miller and several members of his band were injured when their bus overturned on a busy highway in Switzerland, killing the driver.

The German-registered private bus tipped over as it drove into a bend on the A2 highway and came to a rest on its side, police in the canton state of Uri said.

The bus was carrying 13 people - two drivers and 11 members of the Marcus Miller Band.

Over his career, the bassist has worked with jazz greats such as Miles Davis, Dizzy Gillespie and Wayne Shorter, according to his website.

He won two Grammys, his first in 1991 for Best Rhythm & Blues Song for "Power of Love" along with Luther Vandross and Teddy Vann, and the second in 2001 for Best Contemporary Jazz Album for "M2".

The band was on its way from Monte Carlo to the Dutch town of Hengelo, the next stop on the their tour, where it was due to perform tomorrow.

Police spokesman Karl Egli said the 12 passengers were injured and taken to hospitals, but none had life-threatening injuries.

Mr Miller was discharged from the hospital today, as were fellow band members Alex Han and Kris Bowers, but some other band and crew members were being kept in hospitals overnight, according to a post on Mr Miller's official Facebook page.

The cause of the accident was not immediately clear. Police believe no other vehicles were involved.