Company force to serve notice period despite option to buy out period

Hi, I recently resigned from my company after serving for a month. I am currently under probation (which is 6 months in the Company) and Notice period stated in Appointment letter is 2 months during probation. However, I will be able to serve only 3 weeks of the notice period, and have been asking for an early relieving from my present employer since my resigning. They are not budging and asking me to serve the full notice period. I cannot do so, as I have to join an internship program with a new Company, which I already informed them. I have offered to buy-out the period, but they are not agreeing to that as well. Suggest, if my current Company can by-law force me to serve the full notice-period. Though i have stated in email about my last serving day, and to buy-out the notice period, over which my senior manager has verbally threatened me to not give me relieving, spoil my Background check in future and mark my relieving in red, which will consequently ruin my career. Kindly guide.