On my recent visit to the bank, I was told that money coming into the NRE account can only be foreign currency and for NRO accounts, the money can come in local currency but has to be a valid source of income (e.g. rent or investments in India).
Yes this is correct as per FMEA regulation in India.
Now if we use 3rd party remittances like Remitly or Transferwise etc, they usually covert the foreign currency into local currency like INR and then deposit it.
The remittance services are better suited for transferring funds to Normal Savings accounts of your loved ones. Most remittance services would transfer funds using a domestic clearing network [NEFT] and hence the trace that funds originated outside of India is lost. There could be some generic remittance that may have direct tie-up with some banks to do direct transfers.
How can we achieve this in either NRE/NRO accounts? If not, what are the other options ?
You can do a Wire Transfer [SWIFT] from US to Indian NRE account. You can also use the remittance services [if available] from Banks where you hold NRE Account. For example RemittoIndia from HDFC for an NRE account in HDFC, or Money2India from ICICI for an NRE account in ICICI or QuickRemit from SBI etc. These would preserve the history that funds originated from outside India. Similarly you can also deposit a Foreign Currency Check into Indian Bank Account. The funds would take around month or so to get credited.
All other funds can be deposited in NRO account.