If your employees work on multiple projects and require approval by project rather than a single person approving their whole timesheet, you can enable single-project timesheets, allowing you specify approvers for each project.
There are actually 3 modes of timesheets:
Mixed project mode - The employee can add more than one project to their timesheet (if they have been assigned to more than one), and the approver selection is based on the approver assigned to the employee
Single project mode - The employee term can only create timesheets for single projects, however they can create multiple timesheets, each for a different project. They can only add project into the timesheet which belong to the project for which the timesheet has been set up for. If any approvers have been set up for the project, then those approvers will be selected by default. Otherwise the default employee approvers are shown instead.
Mixed mode - The employee can create a mixed project timesheet as well as single project timesheets
Setting up project approvers
If you have chosen Single Project Mode submission, you will need to configure your project approvers. You can do so by editing the project, and clicking on the Approvers tab. You may choose single stage or multiple stage approval for the project.