You're making 6lpa and you (hopefully) don't pay any tax. A lot will change. 13lpa is going to be going to be like 11.75lpa in hand, or say 90k+ per month.
Bangalore is expensive. Nevertheless the growth in income is high, and if you're ready to save a bit (major chunk being in PG) then it'll help a lot.
If you get a PG near office then general expenses are going to be like 25-30k a month to live pretty comfortably.
In Chennai 10.5lpa will get you 80k per month in hand. It's a much better deal to be honest, however you see it.
Sadly enough your negotiating power completely depends on your current salary and competing offers, not on the skillset you bring to the table.
You're better off getting more experience while working in Chennai and then make a leap when you get a 20LPA+ offer, move to Bangalore, and join the TC maximization club.