Hjulmand joined Lecce in the summer of 2021 and went on to play a key role in their promotion from Serie B in his first season, starting 36 of the club's 38 league games.
The Dane, who has been named the club captain, has continued to be a regular feature in the side during their current Serie A campaign, chipping in with four assists in 23 league appearances.
Hjulmand's impressive performances have led to interest from a number of clubs across Europe, including sides from the Premier League.
Southampton had an €11m (£9.7m) bid rejected in the January transfer window, and while it remains to be seen whether the Saints will revisit the transfer, Hjulmand is also being monitored by Leicester.
According to journalist Nicolo Schira, Leicester have sent scouts to watch the Danish international in action.
The Foxes are seemingly keeping tabs on the 23-year-old, as they look to bolster their midfield in the summer transfer window, with Youri Tielemans expected to leave on a free transfer.
The Belgium international is out of contract in June, and while Leicester are keen to retain his services, Tielemans is unlikely to agree to a new deal.
Should Leicester decide to move for Hjulmand, they are expected to face intense competition from several European clubs.
Eintracht Frankfurt and Borussia Dortmund were both previously linked with the Lecce man, while there is also believed to be interest from Juventus, Roma and Atalanta.
With less than 18 months remaining on his current deal, Lecce could be prepared to listen to offers as they look to avoid a situation where he leaves on a free transfer in 2024.
Leicester, meanwhile, have also been linked with the likes of Mahmoud Dahoud and Rani Khedira, who play for Dortmund and Union Berlin respectively.
A recent report also claimed that they had made an enquiry about Inter Milan midfielder Roberto Gagliardini.
After being knocked out of the FA Cup by Blackburn Rovers on Tuesday evening, Leicester will turn their attention to Saturday's Premier League meeting with Southampton.