A tenant may be liable without a signed lease. No signed written lease means an oral lease. An oral lease is a month to month lease, which means it can be changed, going forward, on one month's notice. However, for all the time that the tenant is in the apartment, it is treated for most purposes like a written lease, including as to rent and security deposit.As a general matter, a landlord is not obligated  to let a tenant apply the security deposit to the rent; the landlord can require it be kept intact until the tenancy is up, at which point it may be drawn against for damages or unpaid rent. The landlord also has a right to bring legal action against a tenant who has not paid his rent. So the landlord does not need to let you do this, and can sue if he wants.