Looks like bacterial contamination. I suggest testing everything liquid that you use with these cells. Take small amounts of everything and put them in the incubator for a few days to find out where the contamination is coming from. Also, if you use agar plates to grow your spheroids, test them, too.
Just like Kerry said, aseptically plate an aliquot of your culture, buffers, medium and other reagents onto a non-selective agar (LBE, BHI, nutrient broth or even Müller-Hinton) and incubate it at 37°C, for at least 48 h. Discard all plates/reagents/ bottles that show any colony forming.
Now, the plate my post doc seeded also got contaminated and the only thing I did is take the cover out and put the insert out in another well adjacent and shake it. The funny thing is only the control without the insert is contaminated.
You are welcome Shilpa, but I was not able to help you because I really dont know the exact nature of the problem. When you say: "I have been getting this thread like contamination in my sphere formation." what exactly you mean by "sphere"?