Western Union started their money transfer service in 1871. At the time they were a telegraph company that happened to have this wire transfer business. Today the telegraph is gone but the wire transfers remain as their primary business.
Probably not the absolute earliest but it's a major part of the timeline.
The Federal Reserve in the early 1900s is the earliest I came across in wikipedia, which inspired the question.
https://en.wikipedia.org/wiki/Fedwire
In the early 1900s, settlement of interbank payments was often done by the physical delivery of cash or gold. By 1915, The Federal Reserve Banks began to move funds electronically. In 1918, the Banks established a proprietary telecommunications system to process funds transfers, connecting all 12 Reserve Banks, the Federal Reserve Board and the U.S. Treasury by telegraph using Morse code. Starting in the 1920s up until the 1970s, the system remained largely telegraphic; however as technology improved, they began to make the shift from telegraphy towards telex, then to computer operations and then to proprietary telecommunications networks.[5]
I love how often the answer to “what was the first time someone did X?” turns out to be several times longer ago than one might have imagined (that Ancient Greek battery seems to have surfaced a couple Times this week too).
Probably not the absolute earliest but it's a major part of the timeline.