Due to the importance of continuous monitoring of blood pressure (BP) in controlling hypertension, the topic of cuffless blood pressure (BP) estimation has been widely studied in recent years. A most important approach is to explore the nonlinear mapping between the recorded peripheral signals and the BP values which is usually conducted by deep neural networks. Because of the sequence-based pseudo periodic nature of peripheral signals such as photoplethysmogram (PPG), a proper estimation model needed to be equipped with the 1-dimensional (1-D) and recurrent layers. This, in turn, limits the usage of 2-dimensional (2-D) layers adopted in convolutional neural networks (CNN) for embedding spatial information in the model. In this study, considering the advantage of chaotic approaches, the recurrence characterization of peripheral signals was taken into account by a visual 2-D representation of PPG in phase space through fuzzy recurrence plot (FRP). FRP not only provides a beneficial framework for capturing the spatial properties of input signals but also creates a reliable approach for embedding the pseudo periodic properties to the neural models without using recurrent layers. Moreover, this study proposes a novel deep neural network architecture that combines the morphological features extracted simultaneously from two upgraded 1-D and 2-D CNNs capturing the temporal and spatial dependencies of PPGs in systolic and diastolic BP estimation. The model has been fed with the 1-D PPG sequences and the corresponding 2-D FRPs from two separate routes. The performance of the proposed framework was examined on the well-known public dataset, namely, Multi-Parameter Intelligent in Intensive Care II. Our scheme is analyzed and compared with the literature in terms of the requirements of the standards set by the British Hypertension Society (BHS) and the Association for the Advancement of Medical Instrumentation (AAMI). The proposed model met the AAMI requirements, and it achieved a grade of A as stated by the BHS standard. In addition, its mean absolute errors (MAE) and standard deviation for both systolic and diastolic blood pressure estimations were considerably low, 3.05±5.26 mmHg and 1.58±2.6 mmHg, in turn.