A functor $N\colon\mathrm{Cat}_{A_\infty}\longrightarrow\mathrm{Cat}_\infty$ 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 linear $A_\infty$-categories as special $\infty$-categories?

References

[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