There are two definitions of $A_\infty$-category in common use: linear $A_\infty$-categories and general $A_\infty$-categories.

Regarding the linear definition, a functor $N\colon A_\infty\text{-}\mathrm{Cat}\rightarrow\mathrm{Cat}$ is constructed in a paper [1] by Faonte. This gives a way to get an $\infty$-category by starting with an $A_\infty$-category.

Going the other way, is it possible to define a linear $A_\infty$-category as a special kind of an $\infty$-category?

What about general $A_\infty$-categories (i.e. categories over the $A_\infty$-operad)?

Finally, if such descriptions are possible, where can one read about them?

[1] Simplicial nerve of an A-infinity category (Giovanni Faonte, [arXiv:1312.2127)][2], suggested by DamienC in an answer to [MO152370][1]. [1]: https://mathoverflow.net/questions/152370/are-infty-1-categories-a-infty-categories [2]: https://arxiv.org/abs/1312.2127