I would suggest talking to your doctor
about this so that you can have a diagnosis. If there is no itching or pain in the eyes nor change in vision then it is unlikely to be a serious problem, but you should have it looked at.
From your description, it is hard to know what is going on. However, it sounds like you are seeing an area of bright red pooled blood in the white part of the eye, rather than just having the overall pink color of bloodshot eyes. If this is the case, then this is called a subconjunctival hemorrhage. A subconjunctival hemorrhage occurs when a small blood vessel in the white part of the eye breaks, allowing blood to leak out. Subconjunctival hemorrhages are cosmetically unappealing but they are not dangerous, and they will heal on their own without treatment.
They usually occur due to coughing or straining or sneezing. A nettipot will not help them, per se, however if you have a lot of sinus congestion or nasal allergies, it is possible that the coughing and sneezing associated with these conditions could be the reason that you now have a subconjunctival hemorrhage. Talk to your doctor!