In MSSM models with various boundary conditions for the soft breaking terms ( msoft ) and for a Higgs mass of 126 GeV, there is a (minimal) electroweak fine-tuning Δ≈800 to 1000 for the constrained MSSM and Δ≈500 for non-universal gaugino masses. These values, often regarded as unacceptably large, may indicate a problem of supersymmetry (SUSY) breaking, rather than of SUSY itself. A minimal modification of these models is to lower the SUSY breaking scale in the hidden sector ( f ) to few TeV, which we show to restore naturalness to more acceptable levels Δ≈80 for the most conservative case of low tanβ and ultraviolet boundary conditions as in the constrained MSSM. This is done without introducing additional fields in the visible sector, unlike other models that attempt to reduce Δ . In the present case Δ is reduced due to additional (effective) quartic Higgs couplings proportional to the ratio msoft/f of the visible to the hidden sector SUSY breaking scales. These couplings are generated by the auxiliary component of the goldstino superfield. The model is discussed in the limit its sgoldstino component is integrated out so this superfield is realized non-linearly (hence the name of the model) while the other MSSM superfields are in their linear realization. By increasing the hidden sector scale f one obtains a continuous transition for fine-tuning values, from this model to the usual (gravity mediated) MSSM-like models.