From a domestic perspective, within the US, the form you heard about is likely IRS form 8300, Report of Cash Payments Over $10,000. The requirements for that form include a $10,000 threshold which causes people concern when they're doing bank transactions over that amount. However, the form is strictly targeted and often people find it doesn't apply to them.
Individuals or businesses who are involved in transactions involving more than $10,000 in cash are required to submit that form. As an example, if your neighbor bought a boat from you and he handed you a bag containing $20,000 in cash, you'd need to report it.
However, the form is not required for all transactions over the $10,000 threshold. Firstly, it's not required if the transaction involves a financial institution that's obligated to file CTRs with FinCEN (which your bank almost certainly is). Secondly, it's not required for non-cash transactions. So, unless you're sending the money between banks via currency, check, cashier's check, or some other cash equivalent, you don't need to fill it out. In other words, if you're sending it via electronic transfer or any other non-cash method, you don't need to fill it out.
In some cases, people with bank accounts in the US and elsewhere need to file a FBAR (Foreign Bank Account Report) with FinCEN, but that only applies to residents and citizens of the USA, and it sounds like you're not either of those right now.