The application of computing intelligence in the field of Ophthalmology is gaining particular interest. Retinal fundus image analysis is one such area that is being explored with computational learning techniques. Automatic Identification of left and right eye in fundus images provides insights into the position of various anatomical structures within the retina and also for diagnosis of retinal diseases grounded on properties of the optic disc. The majority of earlier works attempt to identify the left and right eyes through the segmentation of retinal structures. A few works do not involve segmentation but utilize the entire image for this task. This work attempts to identify the left and right eyes without segmenting structures as well as by providing only three-fourths of the image as input. In this regard, the proposed methodology involves resizing the image into lower resolution, transforming it to L*a*b* colour space, applying CLAHE-based contrast enhancement, converting back to the original colour space, and cropping the image such that only three-fourth of the width is retained, scaling image such that pixel values range between 0 and 1, classification model building through a convolutional neural network built from scratch, performance evaluation through validation set of images and evolving of the best learning model for left and right eye classification in fundus images. The proposed methodology is trained and evaluated through a fundus image dataset available in the public repository of Kaggle. The methodology achieves a validation accuracy of 98.63% on 10906 validation images and test accuracy of 98.63% on 42670 test images. Further, the model is fine-tuned to identify left and right eye images in other retinal fundus benchmark datasets namely HRF, DRIVE, Diaretdb0, Diaretdb1, and HEI-MED available in public repositories. With very little fine-tuning with 40% of images from each dataset as training and 60% of images for testing, a classification accuracy of 100% is achieved for all the datasets.