The probability of receiving a helpful reply will be improved if you mention the name of your bank. You will not be their first client who needed to deposit cash.

It appears that Western Union will accept cash and transmit it to a bank account:

https://www.westernunion.com/us/en/s...in-person.html

Scroll down to read: Send money to a bank account

However, the simpler and less expensive solution is to give the cash to a friend or colleague who has a local bank account, and ask the friend to send the same amount to your account using the bank's online bill pay system or to write you a paper check.