Generally a person who is doing the banking business is called banker.
According to Dr.Hart, "A ‘Banker’ is one who in the ordinary course of his business honors cheques drawn upon him by persons from and for whom he received money on current accounts."
According to Sir John Paget, "No person or body corporate or otherwise can be a banker who does not: take deposit accounts. Take current accounts, issue and pay cheques drawn on himself and collect cheque for his customer."
It is said in English Bills of Exchange Act-1882, ‘Banker’ includes a body of persons, whether incorporated or not, who carry on the ‘business of banking’.
So, it can be said, that a banker does 'business of banking' i.e. accepts deposits withdrawable by cheques.