We have considered two distant mesoscopic superconducting quantum interference device (SQUID) rings A and B in the presence of two-mode nonclassical state fields and investigated the correlation of the supercurrents in the two rings using the normalized correlation function CAB. We show that when the parameter α is very small for the separable state with the density matrix (p) = (|α, -α〉 〈α, -α| + |-α, α〉 ( α, α|)/2 and entangled coherent state (ECS)|u〉 = N1(|α,-α〉 +|-α, α〉) fields, the dynamic behaviours of the normalized correlation function CAB are similar, but it is quite different for the entangled coherent state |u’〉 = N2(|α,-α〉 -|-α, α〉) field. When the parameter α is very large, the dynamic behaviours of CAB are almost the same for the separable state, entangled coherent state |u〉 and |u’〉fields. For the two-mode squeezed vacuum state field the maximum of CAB increases monotonically with the squeezing parameter r, and as r → ∞, CAB → 1. This means that the supercurrents in the two rings A and B are quantum mechanically correlated perfectly. It is concluded that not all the quantum correlations in the two-mode nonclassical state field can be transferred to the supercurrents; and the transfer depends on the state of the two-mode nonclassical state field prepared.