Cheapest Bidirectional NFC With Android Phone
You can rig something that seems to communicate both ways with this EEPROM but that would mean writing the EEPROM, then polling for return data.  And since an EEPROM can only be written so many times, this is likely not what you want.

I think you are looking for something like the AMS AS3953A.  From the datasheet, 


  "It is designed to create a fast data link between an ISO 14443A
  reader device (PCD) and a microcontroller."


And:


  Having a NFC Forum compatible tag interface allows
  the AS3953A to be used in an application where a standard NFC
  enabled phone is used as a PCD.


This is available in single quantities from Digikey for a pretty good price.