Takeda Pharmaceutical Co.’s potential pursuit of Shire PLC would rank as one of the biggest-ever attempts by a Japanese company to take over a Western rival, as a strengthening yen helps fuel ambitious overseas expansion.

Takeda said on Wednesday it was weighing a bid for London-listed Shire, whose drugs include Adderall, for attention deficit hyperactivity disorder. The potential deal would create a global drug giant worth more than $80 billion and add to a torrid run for mergers and acquisitions so far this year—much of...