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Published 12:52 22 Jul 2016 BST
Updated 13:30 22 Jul 2016 BST
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The Saints are short of options for the position, following Kelvin Davies' retirement and the end of Maarten Stekelenburg's loan deal. Southampton only have Paulo Gazzaniga as cover for Fraser Forster
The paper report that new manager Claude Puel is looking to sign a new 'keeper, and Randolph is rated highly by the club.
However, to move and be back-up to a goalkeeper who's superior to the one currently keeping him out of the Hammers' team would arguably be a sideways move for the 29-year-old at this stage of his career.
Randolph is currently on West Ham's pre-season tour, and featured in a game in Austria on Tuesday evening.

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