The Sunday People claims that the 49-year-old Dane is likely to step down from his position at the Liberty Stadium in the summer and move onto manage a bigger club.
In the event of such an occurrence, Swans chairman Huw Jenkins has lined up Jones to take charge of the first team.
The 43-year-old, who is currently the assistant manager at Everton, was offered the Swansea job in 2012 when Brendan Rodgers departed, but he did not take it as he did not feel ready.
However, it has now been reported that Jones is prepared to become the manager of the Welsh club next season.
Swansea are currently 12th in the Premier League with 24 points from as many matches.