Approximating solutions of non-linear parametrized physical problems by interpolation presents a major challenge in terms of accuracy. In fact, pointwise interpolation of such solutions is rarely efficient and generally leads to incorrect predictions. To overcome this issue, instead of interpolating solutions directly by a straightforward approach, reduced order models (ROMs) can be efficiently used. To this end, the ITSGM (Interpolation On a Tangent Space of the Grassmann Manifold) is an efficient technique used to interpolate parameterized POD (Proper Orthogonal Decomposition) bases. The temporal dynamics is afterwards determined by the Galerkin projection of the predicted basis onto the high fidelity model. However, such interpolated ROMs based on ITSGM/Galerkin are intrusive, given the fact that their construction requires access to the equations of the underlying high fidelity model. In the present paper a non-intrusive approach (Galerkin free) for the construction of reduced order models is proposed. This method, referred to as Bi-CITSGM (Bi-Calibrated ITSGM) consists of two major steps. First, the untrained spatial and temporal bases are predicted by the ITSGM method and the POD eigenvalues by spline cubic. Then, two orthogonal matrices, determined as analytical solutions of two optimization problems, are introduced in order to calibrate the interpolated bases with their corresponding eigenvalues. Results on the flow problem past a circular cylinder where the parameter of interpolation is the Reynolds number, show that for new untrained Reynolds number values, the developed approach produces sufficiently accurate solutions in a real-computational time.