Galinski, 26, was previously with Grantham Town, Hednesford Town and Alfreton Town and made 34 starts in the 2019/20 season.
His re-signing follows the news that Marcus Marshall has joined Basford from Matlock Town, while midfielder Kyle Dixon returns from Newark Flowserve.
A club statement said: “Whilst great news, given the current pandemic, at a more appropriate time the club will do a fuller announcement.”
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