Sweeney was a product of AFC Wimbledon's youth academy, but was released earlier this year after one league appearance for the club.
Kingstonian snapped him up but, almost since the first day he joined the Ryman Premier League outfit, Football League clubs have been keeping close tabs on him.
Sky Sports understands Ipswich have now made the first move and Sweeney joined them for training this week.
The 19-year-old midfielder, who can also play in defence, will feature for Kingstonian on Saturday, before returning to Suffolk to complete his trial stint.
"Sweeney did really well in pre-season and a number of the coaches thought he had a future in the professional game," a Kingstonian spokesman told Sky Sports.