Following the latest developments, bio-based polyesters, obtained from renewable raw materials, mainly carbohydrates, can be competitive for the fossil-based equivalents in various industries. In particular, the furan containing monomers are valuable alternatives for the synthesis of various new biomaterials, applicable in food additive, pharmaceutical and medical field. The utilization of lipases as biocatalysts for the synthesis of such polymeric compounds can overcome the disadvantages of high temperatures and metal catalysts, used by the chemical route. In this work, the enzymatic synthesis of new copolymers of ε-caprolactone and 5-hydroxymethyl-2-furancarboxylic acid has been investigated, using commercially available immobilized lipases from Candida antarctica B. The reactions were carried out in solvent-less systems, at temperatures up to 80 °C. The structural analysis by MALDI TOF-MS, NMR, and FT-IR spectroscopy confirmed the formation of cyclic and linear oligoesters, with maximal polymerization degree of 24 and narrow molecular weight distribution (dispersity about 1.1). The operational stability of the biocatalyst was explored during several reuses, while thermal analysis (TG and DSC) indicated a lower thermal stability and higher melting point of the new products, compared to the poly(ε-caprolactone) homopolymer. The presence of the heterocyclic structure in the polymeric chain has promoted both the lipase-catalyzed degradation and the microbial degradation. Although, poly(ε-caprolactone) is a valuable biocompatible polymer with important therapeutic applications, some drawbacks such as low hydrophilicity, low melting point, and relatively slow biodegradability impeded its extensive utilization. In this regard the newly synthesized furan-based oligoesters could represent a “green” improvement route.