It seems that the tips of the fan of a turbofan engine can and do break the sound barrier:
The GE-90 has a fan diameter of 3124 mm and a rotational speed of 3475 RPM. Their circumferential velocity is d·π·57.917 = 568 m/s or Mach 1.67 at sea level and ISO atmosphere.
Peter Kämpf's answer
While, on a propeller driven aircraft, it seems that having the prop tips exceed the speed of sound is a bad thing - Can turboprop blades break the sound barrier?
Why is it that exceeding the speed of sound is acceptable in one situation but not the other?