The COVID-19 pandemic situation has created even more difficulties in the quick identification and screening of the COVID-19 patients for the medical specialists. Therefore, a significant study is necessary for detecting COVID-19 cases using an automated diagnosis method, which can aid in controlling the spreading of the virus. In this paper, the study suggests a Deep Convolutional Neural Network-based multi-classification approach (COV-MCNet) using eight different pre-trained architectures such as VGG16, VGG19, ResNet50V2, DenseNet201, InceptionV3, MobileNet, InceptionResNetV2, Xception which are trained and tested on the X-ray images of COVID-19, Normal, Viral Pneumonia, and Bacterial Pneumonia. The results from 3-class (Normal vs. COVID-19 vs. Viral Pneumonia) showed that only the ResNet50V2 model provides the highest classification performance (accuracy: 95.83%, precision: 96.12%, recall: 96.11%, F1-score: 96.11%, specificity: 97.84%) compared to rest of the models. The results from 4-class (Normal vs. COVID-19 vs. Viral Pneumonia vs. Bacterial Pneumonia) demonstrated that the pre-trained model DenseNet201 provides the highest classification performance (accuracy: 92.54%, precision: 93.05%, recall: 92.81%, F1-score: 92.83%, specificity: 97.47%). Notably, the ResNet50V2 (3-class) and DenseNet201 (4-class) models in the proposed COV-MCNet framework showed higher accuracy compared to the rest six models. This indicates that the designed system can produce promising results to detect the COVID-19 cases on the availability of more data. The proposed multi-classification network (COV-MCNet) significantly speeds up the existing radiology-based method, which will be helpful to the medical community and clinical specialists for early diagnosis of the COVID-19 cases during this pandemic.