If a bank is selling the loan it must follow investor requirements.
If the bank will keep the loan in its own portfolio, at present there is the standard requirement that an application be in writing, but that does not necessarily mean written down on a piece of paper by the applicant. It means any application that is:
>written on an application form by the applicant
>taken over the phone or in person by a bank employee and written down on paper or in the bank's system
>an internet application.
It is a common practice, even for applications that will be kept in the bank's portfolio, for a final application to be reviewed and signed at closing. This is particularly customary for loans that will be sold on the secondary market.