Is heating the nichrome wire with phone battery possible?
Would decreasing the wire diameter of nichrome wire to increase the
  resistance help that much to heat up?


This is a tricky question. Temperature rise is proportional to power and inversely proportional to surface area, which is proportional to diameter. So for the same power a thinner wire will get hotter. But (on fixed voltage) power is inversely proportional to resistance, which is inversely proportional to diameter squared. The end result is that a thicker wire will get hotter because the surface area increases less than the power increases.

However this assumes that the voltage is constant. When more current is drawn from a battery its output voltage decreases, due to resistance of its internal connections and electrochemical effects. Assuming the battery has no protection devices to prevent drawing too much current, it will put out maximum power when its internal resistance and the wire resistance are equal (if the wire resistance is lower than the battery resistance it will get less power and won't get as hot, despite higher current draw). But an equal power will also be dissipated inside the battery, making it heat up. 

Batteries are generally not designed to put out maximum power for more than a few seconds at most. If you try to draw maximum power out of a phone battery it may be damaged by the high current draw and/or heat up to a dangerous temperature. The Lithium Ion batteries used in cell phones will explode if they get too hot.      

The other factor affecting wire temperature is length. A shorter length has lower resistance, increasing current draw and making the wire hotter. So a short length of thin wire can get just as hot as a longer length of thick wire, while drawing less current because it needs less power. It will also heat up faster due to its lower thermal mass.


  Do I need any extra circuit to convert the volts into current for
  heating?


That depends on the battery voltage and length of wire you want to heat. The thinner the wire the less current it needs, but if it doesn't draw enough current on the voltage available you will need to boost it. The ultimate limiting factor is the amount of power the battery can safely deliver, which limits the length of any given diameter wire you can heat. 

Calculating the current required to get a particular diameter of Nichrome wire up to the required temperature is also quite tricky, since it is affected by a number of factors including the wire's composition and thermal effects. To get the best effect you may have to experiment with different diameters and lengths (always being careful to not overload the battery!). Here is an interactive online calculator to give you a starting point:-

Jacobs Online: Nichrome Wire Calculator