One wallet that supports the Binance Chain/Binance Smart Chain ecosystem is Trust Wallet. Here is a tutorial that shows how to send BEP2 assets to your Binance.US account from Trust Wallet.
1. Make sure you have BNB to use for fees. Select the BEP2 asset you would like to send.
2. Click on "Send"
3. Retrieve your BEP2 deposit address and memo from the appropriate asset as seen in this image:
4. Enter your Binance.US BEP2 address and DO NOT FORGET THE MEMO. Click on "CONTINUE".
5. On the last screen, ensure that all amounts are correct and that your addresses and memo are correct as well. When you are sure that all is correct, press "SEND"