Dear Anuj,
You are bound to serve a 60-day notice period, possibly due to the terms of separation as laid down in your employment contract. You may be relieved earlier than the 60-day period if the clause for payment in lieu of notice exists in your appointment terms. The consideration of the buyout option is still pending with the employer as the functioning of the establishment would be adversely affected until an arrangement is made.
You should request your manager to reconsider the buyout option. The current situation entirely depends on how you handle it if your manager does not agree to your proposal. Firstly, you need to discuss with the HR department of the new organization:
(i) to extend your joining date by 61 days
(ii) whether they are willing to onboard you without a relieving letter from your current employer.
If they agree to point 1, there should be no issue. In case of disagreement with point 1 but agreement with point 2, you should deposit the 60-day amount with your current employer, but ensure to do so through a lawyer.