The 41-year-old is surplus to requirements at West Ham but is keen to prolong his 23-year professional career.
The Championship side recently lost last season's top scorer Jamie Cureton to Norwich and had been in the market for a striker.
Ex-Manchester United and Tottenham star Sheringham would be the Us' third big signing in 24 hours after they twice broke their transfer record to capture Mark Yeates and Clive Platt.
Tottenham's Irish winger Yeates, 22, was first to arrive.
He spent a spell on loan at Layer Road two seasons ago and had temporary deals at Hull and Leicester in the last campaign.
Striker Platt soon followed for what the club will only say is a "six-figure sum".
Us chief executive Marie Partner told the club's official website: "These are the first two signings in what will be a very busy campaign in securing the club's squad for next season's push in the Championship.
"There is a great deal of exciting news to come."
The Yeates and Platt deals will be formally completed when clearance is received by the Football Association and Football League.