as mentioned by others, the main thing you can negotiate is your entry level in the firm (eg Senior Associate with 12 months to Consultant vs 6 months to Consultant). It would be very difficult to negotiate a different standard salary for a fixed seniority level, as this would create inequality in the firm. On the other hand, if you can negotiate a higher initial entry level, this should bring a higher salary or faster promotion track without creating imbalance in the company, thus would be feasible.
You can definitely try to negotiate your entry level in the firm. The degree of negotiation depends obviously on how valuable you can be for the company (usually the more senior, the more valuable). In any case, you should be aware that with a higher level, expectations on your performance will also be higher.