The 23-year-old has spent his career in the EFL, initially coming through the academy system at Ipswich Town before earning a transfer to Swansea last year.
During his first campaign in South Wales, Downes made a total of 37 league appearances as his club finished in 15th position in the Championship table.
However, while Downes is regarded as a key player at Swansea, club officials may be open to a sale if they can make a substantial profit on a player who cost £1.5m less than 12 months ago.
According to WalesOnline, Downes's price-tag has been set in the region of £11m, a figure which may seemingly tempt a number of top-flight clubs into a formal approach.
Wolverhampton Wanderers, Crystal Palace and Southampton are all said to be monitoring the situation having already made tentative enquires.
Of those Premier League clubs, Wolves are known to be in the market for at least one midfield addition this summer with the West Midlands outfit said to be tracking Sporting Lisbon duo Joao Palhinha and Matheus Nunes.
Nevertheless, at a time when it has been reported that Palhinha will now join Fulham, Wolves have allegedly made little progress in a club-record pursuit of Nunes.
Meanwhile, Crystal Palace require a replacement for Conor Gallagher, who has returned to parent club Chelsea, and head coach Patrick Vieira has shown since his arrival at Selhurst Park that he is prepared to sign younger players who have impressed in the EFL.
Southampton are yet to make any changes to their midfield this summer, but Downes's high-passing success rate last season is a statistic which will catch the attention of boss Ralph Hasenhuttl.
Downes, who earned six caps at Under-20 level for England, still has three years remaining on his contract at the Swansea.com Stadium.
However, the report adds that the sale of either Downes or last season's top goalscorer Joel Piroe may be welcomed to boost the club's financial position.
Furthermore, it is claimed that parting ways with one of the duo would significantly boost their chances of being able to retain the services of the other.
Leeds United have also been credited as admirers of Downes, who may come into their thinking if Kalvin Phillips completes a transfer to Manchester City over the coming days.