In this talk I will report on joint work with Fatemeh Mohammadi and Alfredo Nájera Chávez. For a (multi-)homogeneous ideal inside a polynomial ring we generalize classical Gröbner degenerations to define a multi-parameter family combining all Gröbner degenerations associated with a maximal cone in the Gröbner fan and all its faces. When this construction is applied to the homogeneous coordinate ring of the Grassmannians Gr(2,n) and Gr(3,6) (presented compatible with their cluster structure) the algebra defining the family for a certain maximal Gröbner cone coincides with the universal coefficient cluster algebra. I will review the basic notions needed from Gröbner theory and cluster algebras, explain our general construction and our (computational) results for the Grassmannians.