The transformer is a fundamental building block in deep learning, and the attention mechanism is the transformer's core component. Self-supervised speech representation learning (SSRL) represents a popular use-case for the transformer architecture. Due to transformers' acausal behavior, the use of transformers for SSRL has been predominantly focused on acausal applications. However, several media processing problems, such as speech processing, require real-time solutions. In this paper, we present an implementation of the attention module that enables training of SSRL architectures with low compute and memory requirements, while allowing real-time inference with low and fixed latency. The attention module proposed in this paper includes two components, streaming attention (SA) and low-latency streaming attention (LLSA). The SA represents our proposal for an efficient streaming SSRL implementation, while the LLSA solves the latency build-up problem of other streaming attention architectures, such as the masked acausal attention (MAA), guaranteeing a latency equal to one layer even when multiple layers are stacked. We present a comparative analysis between the vanilla attention, which we will refer here as acausal attention (AA), the SA, and the LLSA, by training a streaming SSRL with automatic speech recognition as downstream task. When training on librispeech-clean-100 and testing on librispeech-test-clean, our low-latency attention module has a word error rate (WER) of 5.84%, which represents a significant improvement over the MAA (WER = 13.82%). Our implementation also reduces the inference latency from 1.92 to 0.16 seconds. The proposed low-latency module preserves many of the benefits of conventional acausal transformers, but also enables latency characteristics that make it applicable to real-time streaming applications.