An important aspect of SVM-based speaker verification systems is the design of sequence kernels. These kernels should be able to map variable-length observation sequences to fixed-size supervectors that capture the dynamic characteristics of speech utterances and allow speakers to be easily distinguished. Most existing kernels in SVM speaker verification are obtained by assuming a specific form for the similarity function of supervectors. This paper relaxes this assumption to derive a new general kernel. The kernel function is general in that it is a linear combination of any kernels belonging to the reproducing kernel Hilbert space. The combination weights are obtained by optimizing the ability of a discriminant function to separate a target speaker from impostors using either regression analysis or SVM training. The idea was applied to both low- and high-level speaker verification. In both cases, results show that the proposed kernels outperform the state-of-the-art sequence kernels. Further performance enhancement was also observed when the high-level scores were combined with acoustic scores.