This from OED:
Etymology: German, < Italian giro circulation (of money) A system whereby credits are transferred between banks, post offices, etc.; spec. a system operated by the British Post Office for the banking and transfer of money. Frequently attrib.
1896 P. des Essars in Hist. Banking III. ix. 384 Germany has a slight advantage arising from the fact that in that country cheques and transfers (giro accounts) are more in use than in France and Belgium.
“Giro” (sense 1, a circuit, related to gyrate) is attested as far back as 1670.