Is there any Ethernet adapter to be used with the USB interface of the Mega 2560 board ? I am aware that there are Ethernet boards to be fixed on top of the mega boards. But, if I want to have just an adapter for Ethernet which can be connected to the USB interface of the Mega board. Someone can please help ?
No, sorry, that is not possible. The Arduino Mega 2560 is not a computer, it is a microcontroller.
There is a USB host shield, but an Ethernet adapter is not supported I think : https://www.arduino.cc/en/Main/ArduinoUSBHostShield
The Raspberry Pi is a mini computer, that might support an ethernet adapter, but it has Ethernet already.
The Arduino Yun has a module that runs linux with OpenWRT. If OpenWRT supports the adapter, you might make that work, but I doubt if it will work, and the Arduino Yun already has Ethernet and wifi.