This research work focuses on a Near-Infra-Red (NIR) finger-images-based multimodal biometric system based on Finger Texture and Finger Vein biometrics. The individual results of the biometric characteristics are fused using a fuzzy system, and the final identification result is achieved. Experiments are performed for three different databases, i.e., the Near-Infra-Red Hand Images (NIRHI), Hong Kong Polytechnic University (HKPU) and University of Twente Finger Vein Pattern (UTFVP) databases. First, the Finger Texture biometric employs an efficient texture feature extracting algorithm, i.e., Linear Binary Pattern. Then, the classification is performed using Support Vector Machine, a proven machine learning classification algorithm. Second, the transfer learning of pre-trained convolutional neural networks (CNNs) is performed for the Finger Vein biometric, employing two approaches. The three selected CNNs are AlexNet, VGG16 and VGG19. In Approach 1, before feeding the images for the training of the CNN, the necessary preprocessing of NIR images is performed. In Approach 2, before the pre-processing step, image intensity optimization is also employed to regularize the image intensity. NIRHI outperforms HKPU and UTFVP for both of the modalities of focus, in a unimodal setup as well as in a multimodal one. The proposed multimodal biometric system demonstrates a better overall identification accuracy of 99.62% in comparison with 99.51% and 99.50% reported in the recent state-of-the-art systems.