What you are looking for may be similar to something I queried Ross Street in regard to earlier this summer. I'll save him some time by putting here the relevant part of his response.
I think the one you first mention is what we have been calling V-actegories. Benabou looked at these rather than (as well as?) V-categories in the early days of monoidal categories. Pareigis also made use of them. More recently, publications of Paddy McCrudden involve them. There is a close connection with V-categories. A V-module V x A --> A in this sense for which we have a parametrized adjoint V(x,[a,b]) =~ A(x.a,b) makes A a V-category with V-valued hom [a,b].
Conversely, a tensored V-category becomes such a V-module.
I recommend the work of McCrudden, who has developed among other things a descent theoretic approach to the tensor product of V-actegories. There is also resource in the work of Harald Lindner. His paper, Enriched Categories and Enriched Modules, in Cahiers, Vol XXII-2 (1981) develops morphisms between enriched categories and actegories, which he calls modules. I'm curious about why it is that I have never seen his work referenced. Paul B Levy writes:
Hi
Is there a standard reference for the notion of "left module for a category"? (or right module, or bimodule)
Is there any reference in the setting of ordinary categories rather than (or as well as) enriched categories or bicategories?
Thanks Paul