It's difficult to assess the best strategy without knowing how it's all set up, but a write hook could be useful if the external table has a link to the manager. On employee form it would look something like this:

def write(self, vals):
    for rec in self:
        if rec.<integer_field> == 2 and vals.get('<integer_field>') == 3:
            rec.employee_id.manager_id.<let_manager_know>()
    return super(<ManagerClass>, self).write(vals) 

Generally it's easier to check for a change at the source unless it's for a computed field, in which case a depends is your best bet.