Apparently, your CI shows that 0 is a compatible value. How would you judge this from a practical view? Likewise, I do not know anything about your variables, but you should always check not only if 0 is compatible, but how compatible it your CI with the smallest effect size of interest (SESOI). You may look for this term to get an impression. Even small effects, although statistically significant, may be of no practical relevance.
I wonder if you can increase the number of bootstrap replicates or increase the decimal places on the output. Not that 0.001 or - 0.001 is much different from 0, but at least it would give you clear answer.
Also, what effect size statistic are you using ? For most standardized effect size statistics a value of 0.007 or 0.065 isn't that far from 0 anyway. So, even if significantly different from 0, it might just be that the effect size is too small to be of any practical importance.
What type of bootstrap CI? I ask because, say I am wanting to know the mean number of apples eaten yesterday by people and my sample is:
0,0,0,2,5
Then if just using the middle 95% of resamples from these it will touch 0. BUT, the test for the mean number of apples being equal to 0 in the population can be rejected because of the value 2 (and again the value 5). More details are necessary to provide a useful response imo.