The current option really is to only allow line managers to approve time sheets, however line managers do not equal project managers and therefore line managers do not fully understand the hours booked to a project (they just want the time sheet to be submitted!).
Problem
User adds time to project, line manager approves their time sheet (because they don't know the details of the project). Project manager reviews time sheets / budgets and notices someone has incorrectly book hours (or too many hours) to their project, they then have to move those hours to tidy up their project - time consuming.
Solution
Add a new option which means that project managers approves their 'elements' of someone's time sheet based on the hours booked to their projects. If someone works on several projects, then it may need multiple project managers to approve their time sheet (and probably then their line manager to approve non-project related work). Basically this is adding additional approval processes to the time sheet process, I've work on systems that do this and every project manager whom is responsible for their budgets must be screaming for this control.
