The 21-year-old German-born stopper has joined the Pilgrims following spells with Torquay United and Alfreton Town earlier in the campaign.
He made eighteen appearances for the Gulls before appearing once for the Reds (due to an injury to Matt Duke) after relocating north once again.
Spiess sprang to prominence as a teenager at Notts County and made nine senior appearances during his time at Meadow Lane.
He has also enjoyed loan spells with the likes of Corby Town and Bristol Rovers over recent seasons.
Subject to his signing being ratified by the National League and the Football Association, Spiess will be eligible to make his United debut at Curzon Ashton on Saturday.
Craig Singleton