The former England striker has reportedly been given the go-ahead by West Ham, where he is attacking coach, to speak to the League Two club.
Borough declined to comment on the reports, but Graham Westley’s contract as manager expires imminently and he has not received an offer of an extension.
The move would be 49-year-old Sheringham’s first into management after his appointment to become a member of the Hammers’ staff 12 months ago.
Sheringham started his career with Millwall and played for Nottingham Forest, Tottenham (twice), Manchester United, Portsmouth, West Ham and Colchester before retiring aged 42. He earned 51 caps for England.
Westley suggested after the play-off loss to Southend last week that he may not be at Broadhall Way next season.
He said: “We will see where things go from here. I am the Stevenage manager up until the end of May.
“There’s been no offer to me. The club is having a serious rethink about what it’s going to do for the future so we’ll see how that pans out.”