Problem reading an EEPROM chip using I2C protocol

I am trying to read an EEPROM chip that supports the I2C protocol (cannot tell the IC model number as it is not printed). For sure it supports the I2C protocol, since I wrote a code to detect it using the I2C library and the device address I got it return is 0x51. Now I am trying to write a code that reads data from this IC chip. The code is as follows:

#include <Wire.h>

int addr = 0;

void setup() 
{
    Wire.begin(); // initialise the connection
    Serial.begin(9600);
    while (!Serial) 
    {
    }
    delay(100);
}

void loop() {
  byte deviceAddress = 0x51;
  byte data = readData(addr, deviceAddress);
  Serial.print(data, HEX);
  Serial.print(" ");
  addr++;

  if(addr%16 == 0) {
     Serial.print('\n');
  }
  // check for 1Kbits first
  if (addr%128 == 0) {
     Serial.println("round complete");
     Serial.println();
     addr = 0;
  }
  delay(100);
}

byte readData(int address, int deviceAddress) {
  // sending device address
  Wire.beginTransmission(deviceAddress);
  Wire.write(address);
  Wire.endTransmission();
  Wire.requestFrom((short int)deviceAddress, 1);
  if(Wire.available()) 
  {
    byte data = Wire.read();
    return data;  
  }
  return 0xAA; // random data
}

The problem I am facing is, I get back the address (from which I want to read the data) as the data itself (For e.g. read(0) returns 0, read(1) returns 1 and so on). I even tried to debug the I2C communication using the logic analyzer (Saleae logic in this case). The screenshot attached is shown below.

The screenshot shows the logic for a read operation from a single address(0x78), but the story holds for each and every address i.e. I get back the address instead of the data from the address.

The output of the above code is as follows:

0 1 2 3 4 5 6 7 8 9 A B C D E F
10 11 12 13 14 15 16 17 18 19 1A 1B 1C 1D 1E 1F
20 21 22 23 24 25 26 27 28 29 2A 2B 2C 2D 2E 2F
30 31 32 33 34 35 36 37 38 39 3A 3B 3C 3D 3E 3F
40 41 42 43 44 45 46 47 48 49 4A 4B 4C 4D 4E 4F
50 51 52 53 54 55 56 57 58 59 5A 5B 5C 5D 5E 5F
60 61 62 63 64 65 66 67 68 69 6A 6B 6C 6D 6E 6F
70 71 72 73 74 75 76 77 78 79 7A 7B 7C 7D 7E 7F
round complete

Can you help me by identifying what possibly I am doing wrong here?

Thanks