I'm not sure about the case of general monoidal categories. (Although I seem to recall a remark that there is no such structure since the category of monoidal categories is not cocomplete and a suitable replacement would be the category of multicategories. Perhaps somebody can confirm this.)

However, the case of strict monoidal categories is known since they are simply monoids internal to the category of categories. It is a theorem of Schwede and Shipley (Theorem 4.1 in *Algebras and Modules in Monoidal Model Categories*) that if a monoidal model category satisfies so called *monoid axiom*, then the category of monoids in it inherits a model structure. The monoid axiom is a technical condition which is, in particular, automatically satisfied if all objects are cofibrant as is the case in the canonical model structure on the category of categories.