Sir W. R. Hamilton investigated the real quaternions as an extension to the complex numbers in the middle of the 19th century. This investigation was only one of the his contributions made to mathematical science, it was not the last one. Sir W. R. Hamilton also discovered the complex quaternions. About a century after this discovery, A. F. Horadam introduced the complex Fibonacci numbers and Fibonacci quaternions. Then, by inspiring from these definitions, S. Halici defined the complex Fibonacci quaternions and discussed some properties of them. Recently, the elliptic biquaternions, which are generalized form of the complex and real quaternions, have been presented. In this study, we introduce the set of Fibonacci elliptic biquaternions that includes the set of complex Fibonacci quaternions as a special case, and investigate some properties of Fibonacci elliptic biquaternions. Furthermore, we give the Binet formula and Cassini identity in terms of Fibonacci elliptic biquaternions. Finally, we obtain elliptic and real matrix representations of Fibonacci elliptic biquaternions.