  We consider an extension of the instanton bundles definition, given by
Casnati-Coskun-Genk-Malaspina, for Fano threefolds, in order to include non
locally-free ones on the blow-up $\widetilde{\mathbb{P}^{3}},$ of the
projective $3-$space at a point. With the proposed definition, we prove that
any reflexive instanton sheaf must be locally free, and that the strictly
torsion free instanton sheaves have singularities of pure dimension $1.$ We
construct examples and study their $\mu-$stability. Furthermore, these sheaves
will play a role in (partially) compactifying the t'Hooft component of the
moduli space of instantons, on $\widetilde{\mathbb{P}^{3}}.$ Finally, examples
of these are shown to be smooth and smoothable.
