The metrizability of sprays, particularly symmetric linear connections, is studied in terms of semi-basic 1-forms using the tools developed by Bucataru and Dahl in \cite{b:d}. We introduce a type of metrizability in relationship with the Finsler and projective metrizability. The Lagrangian corresponding to the Finsler metrizability, as well as the Bucataru--Dahl characterization of Finsler and projective metrizability are expressed by means of the Courant structure on the big tangent bundle of $TM$. A byproduct of our computations is that a flat Riemannian metric, or generally an R-flat Finslerian spray, yields two complementary, but not orthogonally, Dirac structures on $T^{\text{big}}TM$. These Dirac structures are also Lagrangian subbundles with respect to the natural almost symplectic structure of $T^{\text{big}}TM$.