I do not think that "linearity" of the phase response is a criterion for a filter to be "minimum-phase".

Filters with a minimum-phase characteristic do not have zeros in the right half of the s-plane (RHP) - and therefore, Butterworth filters are minimum-phase filters (no hidden allpass parts).

Thank you very much...

As Lutz said, a continuous-time system is minimum phase if all zeroes are on the LH half plane. For discrete system all zeroes will be inside the unit circle and the system will also have minimum group delay. Butterworth digital filters dohave all zeroes inside the unit circle, so they are 'minimum phase'.

The Butterworth refers to filters which have a certain Butterworth pole polynomial. With various transformations they can be low-pass,high-pass,bandpass or bandstop. I assume here you are referring to the low-pass prototype type. From this you apply various transforms (lowpass to lowpass, lowpass to highpass etc to get the filter you wish) A continuous time low-pass Butterworth filter has no zeros at all, only poles. Hence it is minimum phase.

"A continuous time low-pass Butterworth filter has no zeros at all, only poles. "

Yes - that`s the reason it belongs to the class of "allpole filters".

The discrete time Butterworth low-pass does have zeroes.

A rational function of a complex variable (either in s or in z for analog or digital filters) has as many zeros as poles (see holomorphic functions of a complex variable). Consider the lowpass case as follows. In the analog filters the standard Butterworth filter type has a constant numerator and a polynomial (in s) denominator. Thus it has a pole distribution dictated by the denominator zeros, while all its zeros are located at infinite frequency and they are equal in number to the degree of the denominator. The corresponding digital filter (for example derived from a bilinear transformation or in some direct design procedure-see my early papers on the subject downloadable from ResearchGate) has similarly all its poles dictated by its denominator located inside the unit circle whilst all its zeros are located at -1.

Exact phase linearity and minimum phase requirements are incompatible. As a demonstration, an FIR digital filter with linear phase requires its impulse response to be symmetric about its mid-point (ie palindromic) and hence for every zero there exists its corresponding inverse. Normally in practice if both requirements are needed then a balance between these is sought through a design procedure based on optimization.