This earthquake is a so-called orphan. There are records in Japan for Tsunamis waves arriving on the Sanriku coast, Japan in 1586. Initially an earthquake in Peru in the same year was considered as its offspring. However directionality from Peruvian tsunamis is not a source of Japanese orphan tsunamis. Butler et al. (2017) propose a megathrust earthquake of Mw>9.25 along the Aleutian islands which would also fit paleo-tsunami records from Northern Hawaii.
Satake et al. (2020) also support that the 1586 Peru earthquake is probably not linked to a records from Japan (and even doubt the records themselves). Witter et al. (2019) question that the Aleutian Islands tsunami took place in 1586, they propose dates earlier in the same century which.
Despite the uncertainties of the Peruvian tsunami in this matter, there is still evidence for a paleo-tsunami in these years leaving traces in Hawaii.

Epicentre for this earthquake was estimated by the authors of this database.