Grassmannian$\mathcal {G}_{q}(n,k)$is the set of all$k$-dimensional subspaces of the vector space$\mathbb {F}_{q}^{n}$. Recently, Etzion and Zhang introduced a new notion called covering Grassmannian code which can be used in network coding solutions for generalized combination networks. An$\alpha $-$(n,k,\delta)_{q}^{c}$covering Grassmannian code$\mathcal {C}$is a subset of$\mathcal {G}_{q}(n,k)$such that every set of$\alpha $codewords of$\mathcal {C}$spans a subspace of dimension at least$\delta +k$in$\mathbb {F}_{q}^{n}$. In this paper, we derive new upper and lower bounds on the size of covering Grassmannian codes. These bounds improve and extend the parameter range of known bounds.