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One of the most popular and widely used techniques for analyzing remotely sensed hyperspectral data is spectral unmixing, which relies on two stages: (i) identification of pure spectral signatures (endmembers) in the data, and (ii) estimation of the abundance of each endmember in each (possibly mixed) pixel. Due to the high dimensionality of the hyperspectral data, spectral unmixing is a very time-consuming task. With recent advances in reconfigurable computing, especially using field programmable gate arrays (FPGAs), hyperspectral image processing algorithms can now be accelerated for on-board exploitation using compact hardware components with small size and cost. Although in previous work several efforts have been directed towards FPGA implementation of endmember extraction algorithms, the abundance estimation step has received comparatively much less attention. In this work, we develop a parallel FPGA-based design of the image space reconstruction algorithm (ISRA), a technique for solving linear inverse problems with positive constraints that has been used to estimate the abundance of each endmember in each pixel of a hyperspectral image. It is an iterative algorithm that guarantees convergence (after a certain number of iterations) and positive values in the results of the abundances (an important consideration in unmixing applications). Our system includes a direct memory access (DMA) module and implements a pre-fetching technique to hide the latency of the input/output communications. The method has been implemented on a Virtex-4 XC4VFX60 FPGA (a model that is similar to radiation-hardened FPGAs certified for space operation) and tested using real hyperspectral data sets collected by the Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) over the Cuprite mining district in Nevada and the Jasper Ridge Biological Preserve in California. Experimental results demonstrate that our hardware version can significantly outperform an equivalent software version,- thus being able to provide abundance estimation results in near real-time, which makes our reconfigurable system appealing for on-board hyperspectral data processing.