Resignation accepted and relieving yet to be decided - new employer not ready to extend my joining date

I have put my resignation on 20.05.2019 and asked my organization to relieve me by 21.06.2019. As per contract agreement, my notice is three months or surrender leaves either of. Since our present organization is under transition. I had discussed with all possible ppl up the ladder. They are not giving my relieving date and asking me to serve the notice of 3 months as per contract. I told I am ready to surrender my leaves but still they are not agreeing.
My new employer where I have to join I have explained the situation they are not ready to extend my joining date which during striking the deal have tentatively agreed upon as 15.07. 2019. Kindly help me how to deal.

Before agreeing the date of Joining you would have consulted, one should always read the termination clause of an appointment letter, if it states that you have to serve 3 months notice then you have to, there is no other option. Relieving you with salary in lieu of notice period is with the hands of your HOD/Management they may grant or may refuse there is no compulsion to accept what the employee says, the choice is always with the employer.
Now the only way out is to either convince your HOD to relieve you at the earliest possible i.e. before July 15, 2019 or take back the resignation as your new employer is reluctant to extent the joining date