Notice must be in writing - not verbal. So you have not received proper legal notice.
Notice must be at least 28 days. Since the tenant is wanting to give notice, they are liable for rent until the last day of the rental period (month of July).
If your tenant is to give you proper written notice anytime between today and July 3rd, the tenant is responsible for the full months rent. They can move out early, if they choose, but still have to pay for the full month.
704.19 Notice necessary to terminate periodic tenancies and tenancies at will
(3) Length of notice. At least 28 days' notice must be given except in the following cases: If rent is payable on a basis less than monthly, notice at least equal to the rent-paying period is sufficient; all agricultural tenancies from year-to-year require at least 90 days' notice.
(4) Contents of notice. Notice must be in writing, formal or informal, and substantially inform the other party to the landlord-tenant relation of the intent to terminate the tenancy and the date of termination. A notice is not invalid because of errors in the notice which do not mislead, including omission of the name of one of several landlords or tenants.
(5) Effect of inaccurate termination date in notice. If a notice provides that a periodic tenancy is to terminate on the first day of a succeeding rental period rather than the last day of a rental period, and the notice was given in sufficient time to terminate the tenancy at the end of the rental period, the notice is valid; if the notice was given by the tenant, the landlord may require the tenant to remove on the last day of the rental period, but if the notice was given by the landlord the tenant may remove on the last day specified in the notice. If a notice specified any other inaccurate termination date, because it does not allow the length of time required under sub. (3) or because it does not correspond to the end of a rental period in the case of a periodic tenancy, the notice is valid but not effective until the first date which could have been properly specified in such notice subsequent to the date specified in the notice, but the party to whom the notice is given may elect to treat the date specified in the notice as the legally effective date. If a notice by a tenant fails to specify any termination date, the notice is valid but not effective until the first date which could have been properly specified in such notice as of the date the notice is given.