Evaluation of the primary inductance of a transformer whose core is saturated
When an inductor is not saturating, the inductance is constant because magnetic permeability is constant. This applies when applying an AC waveform or applying a moderate AC waveform superimposed on some moderate level of DC current. If the core is not saturating the inductance measured (or predicted) is constant.

If the core is saturating then there are two scenarios to consider: -


The AC excitation is producing peaks in the magnetic flux that are flattened out due to core saturation
The AC excitation is small but there is a superimposed DC current that is causing saturation positively or negatively.


In the first of those scenarios, it makes no sense to calculate (or estimate) inductance because inductance (as a numerical value) has no real meaning anymore. It has no real meaning because it is changing throughout the cycle of the applied AC waveform because permeability is changing.

In the 2nd scenario there is some merit in estimating inductance because the AC excitation is "moderate" and, it may be assumed that at its biased point on the BH curve, the permeability is fairly constant. This sort of circuit can be used to re-tune oscillators and also be used in what were called "magnetic amplifiers" (or saturable reactors).


  I would like to know if it is possible to theoretically estimate
  reasonably the primary inductance of a transformer when its core is
  saturated ?


The most practical and certain method is to look at the data sheet of the core material and extract the BH curve. Then, using the DC current value and number of turns and core dimension \$\ell\$, calculate H (magnetic field strength) and position that H value on the BH curve and extract the permeability value. Using other core dimensions you can extract inductance.