RE: Is it possible to assign timesheet approvers to jobs?
Happy New Year,
When setting up Timesheets and their approval, an Approver can be 1) Timesheet Approver on Resource Card 2) Person Responsible on the Job Card. Note that the Resource Setup helps defines this. In the field Time Sheet by Job Approval if you select Never, it goes to the Resource Card Approver. If you select Always, it moves to the Job Card Person Responsible. Machine Only creates a fork in the road - If the machine timesheet relates to a Job, it goes to Person Responsible on Job Card, if linked to a Resource, it goes to Approve on Resource Card.

Update your setup and if the PM is the Person Responsible on the Job Card, they will be the approver of time.

Hope this helps and answers your question.
Thanks,
Steve