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.

Dozens were also injured during the event, in Rapoosal, in the southern state of Tamil Nadu.
The sport, known as jallikattu, was banned by India's Supreme Court in 2014.
But on Saturday the central government temporarily lifted the ban, after large-scale protests.
Several villages and towns in Tamil Nadu then held events on Sunday attracting huge crowds.
Subduing angry bulls has long been practised in the state as a sport and is a key part of the harvest festival.
The court had banned jallikattu on the grounds that it was cruel to animals.
Animal rights activists say the spectacle causes unnecessary stress to the bulls who are released into a crowd and forced to fend off people trying to ride them.
Many in Tamil Nadu, however, are against the ban as they see jallikattu as an important part of their cultural heritage, and also say it ensures the preservation of native breeds of bulls.
In recent days large demonstrations have been staged in the state capital Chennai (Madras) calling for the practice to be fully re-legalised.
Most ministers in the state government including Chief Minister O Panneerselvam, as well as Tamil celebrities such as Oscar-winning composer A R Rahman, support jallikattu.

Dujardin, 30, and Valegro won individual and team dressage gold for Britain at London 2012 and have since won World and European titles.
But, she says, the Olympics in Brazil next summer will be the horse's last.
"This will be Valegro's retirement after Rio so I want to go out there and want to enjoy every last minute," Dujardin told BBC Points West.
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Dujardin partnered Valegro to victory in London as the two performed to movie themes including The Great Escape and Live And Let Die.
And the Gloucestershire rider says she is hugely excited at the prospect of defending her title, while admitting the Games in Brazil represents a huge logistical challenge.
"He's got to stay fit and healthy and get ready for the flight," she said. "It's a 16-hour flight and he needs to acclimatise.
"I'm really, really excited heading towards Rio. We get to go out in February and have a look around and fingers crossed it all goes well.
"Hopefully it won't be too hot, it's their winter time so high 20s."
Dujardin also says she will not let the pressure of being Olympic champion affect her.
"I just take it all on board," she said. "I try not to let that all bother me. Riding Valegro always makes you smile so I enjoy it."
Former England striker Teddy Sheringham held talks with Swindon Town on Thursday about the managerial vacancy at the League Two club.
[NEXT_CONCEPT]
Two men have been gored to death in a bull-wrestling festival, as a fierce debate rages over the controversial sport.
[NEXT_CONCEPT]
Double Olympic champion Charlotte Dujardin says her horse Valegro will retire after the Rio Games in 2016.