Offline timesheet selects the internal e-mail address set for a user, by default. However, the user also gets an option to enter an arbitrary e-mail address which is possible if the necessary permissions are provided.

Please follow the below instructions to enable arbitrary e-mail address option for offline timesheet.
 
1] Log into Web TimeSheet with administrative rights. 
2] Click on 'Administration' icon at the top. 
3] On the left side pane, under 'Permissions' click on edit button on the required permission.

4] Expand the TimeSheet link and enable "Allow entry of arbitrary e-mail address for Offline TimeSheet " option.
5] Now scroll down and click 'Save'.