By 'negative covariance' you probably meant to say 'negative variance'? A commonly used approach to deal with this is to treat it as zero. You may refit the model without the corresponding random effect in the model.

Negative variance is often due to large differences among the data obtained from different replications. So minimizing the soil heterogeneity for all replications is of utmost importance. In your case as you have already taken the data so you better consider it zero 

Thanks for your replies. 

It's probably not significantly less than zero.  Note that under the null hypothesis that genetic variance is zero, the distribution of the estimates should be symmetric around zero so that 50% of the time the estimate will be negative.  The likelihood ratio test (LRT) statistic (minus twice the difference between the log-likelihood of the model with genetic variance and that where the variance is fixed to zero) is asymptotically distributed as chi-squared with 1df.  In models where the estimate is prevented from being less than zero, the distribution is a 50:50 mixture of zero and chi-squared with 1df.  I mention this because it is not so widely appreciated that tests for genetic variance are typically 1-tailed so the LRT p-value should be halved.  In your case, noting that there is no evidence of genetic variance should be sufficient.

I have also got the same negative heritability (-3%) for Harveat Index in pooled analysis for RILs. But interestingly the heritability for harvest index was  high in individual year (75% and 91%). My design was alpha lattice and 300 RILs were evaluated for heat tolerance in Chickpea. Analysis was done in SAS proc Mixed model.

Could you please tell me why I got these results and if it is okay then how to interpret the results?