The former Tottenham and Manchester United forward is one of several to have met officials from the club.
Sheringham, 51, had a spell in charge of League Two Stevenage in 2015-16 but was sacked after 33 games.
Swindon will start next season in League Two having been relegated last term after winning only 11 of 46 games.
Sheringham has been out of management since leaving Stevenage - his first managerial job - in February 2016 after winning just seven of his games as boss.
Previous Robins boss Luke Williams left his role soon after the club's relegation was confirmed, having won two of their final 11 league matches with him in charge.
Since then, Swindon have had an approach for Mansfield boss Steve Evans turned down, and on 23 May chairman Lee Power said although the process had taken longer than expected they had a three-person shortlist.
As part of structural changes at the County Ground, director of football Tim Sherwood's role appears likely to be scrapped.
Swindon won just seven of 31 games following former Tottenham and Aston Villa boss Sherwood's arrival in November 2016.
Former England striker Teddy Sheringham held talks with Swindon Town on Thursday about the managerial vacancy at the League Two club.