This work concerns the production of fibrous composite materials based on biodegradable polymers such as alginate, dibutyryl chitin (DBC) and poly-ε-caprolactone (PCL). For the production of fibres from these polymers, various spinning methods were used in order to obtain composite materials of different composition and structure. In the case of alginate fibres containing the nanoadditive tricalcium phosphate (TCP), the traditional method of forming fibres wet from solution was used. However in the case of the other two polymers the electrospinning method was used. Two model systems were tested for biocompatibility. The physicochemical and basic biological tests carried out show that the submicron fibres produced using PCL and DBC have good biocompatibility. The proposed hybrid systems composed of micrometric fibres (zinc and calcium alginates containing TCP) and submicron fibres (DBC and PCL) meet the requirements of regenerative medicine. The biomimetic fibre system, the presence of TCP nanoadditive, and the use of polymers with different resorption times provide a framework with specific properties on which bone cells are able to settle and proliferate.