The Tele understands that Well youth coach Jonatan Johansson was impressed with Fulton during his brief spell as reserve coach at Ton and has offered the player a trial at Fir Park as a result.
Local lad Fulton joined the Cappielow club in the summer of 2011 after his release from St Mirren.
The left midfielder, who also spent several years with the Celtic youth academy, shone in Mick Rankin's under-19s last season and soon earned a spot on the first-team bench.
Although Fulton went unused in home draws against Hamilton and Ross County, his impressive youth-team displays saw him make the step up to David Hopkin's reserve side this term.
He made a slow start to life there, with Ton boss Allan Moore telling the Tele in October that his application wasn't all that it could be.
He said: "We have actually been giving Aidan a little bit of stick as we think that he has all the ability in the world but his attitude wasn't right.
"We told him he was in the last chance saloon, but the boy, according to David Hopkin, has come in and put in a good shift in training."
The 19-year-old then went on to play an important part in the second string's surge to the top of the table, scoring against Rangers and Queen's Park.
Meanwhile, Morton reserves will face Port Glasgow Juniors in a friendly at Parklea on Wednesday night. The match kicks off at 7pm is £4 entry.