The 22-year-old moved to Bramall Lane from Norwegian outfit Sarpsborg 08 in September 2020 for a small fee, but has since failed to make a massive impact in Yorkshire.
Coulibaly spent the first two seasons on loan at the Blades' sister club Beerschot in the Belgian Pro League before playing a bit-part role for Paul Heckingbottom's men last term.
The Mali international featured just once in the entirety of the Championship campaign as United gained automatic promotion to the Premier League, although he did make five appearances in the FA Cup, including a cameo off the bench at Wembley against Pep Guardiola's Manchester City in the semi-final.
According to Swedish news outlet Fotboll Direkt, a club operating in the Allsvenskan is keen on bringing Coulibaly back to Scandinavian after over three years away.
The report claims that AIK are interested in securing the services of the midfielder, with the Solna-based club looking to stregthen their ranks after a disappointing 11th-placed finish in the top flight in 2023.
It is understood that Chris Wilder and Sheffield United would not stand in the way of a switch elsewhere for Coulibaly, meaning that a New Year transfer is looking increasingly likely.
The one-time international is said to be keen on a move away from Bramall Lane and the Premier League, as the youngster seeks regular first-team action.
Coulibaly's last 12 months with the Blades has been plagued by a recurring knee injury which saw him undergo surgery earlier in the calendar year, before swelling to the area forced him to miss part of the team's pre-season camp in the summer.
A second-round EFL Cup tie against third-tier Lincoln City is the only competitive action that the midfielder has enjoyed in 2023-24 - a game which the top-flight side lost on penalties.
Although a move to Northern Europe is being touted for Coulibaly, a number of other sides within the continent have been credited with a fondness of the player in the past.
It is believed that Turkish giants Galatasaray were prepared to pay £12.9m for the Mali international during 2021, while Serie A's AC Milan have also been interested over recent years.
Possible moves to San Siro and Rams Global Stadium did not materialise however, and Coulibaly is now set for a switch back to Scandinavian football.
Sheffield United have collected three points from their first three matches under the stewardship of Wilder, who takes his Blades side to title-chasing Aston Villa on Friday night.