The 20-year-old centre-back came through the youth ranks at Old Trafford and has played for Scotland Under-19s.
Ahead of Sunday’s home William Hill Premiership game against Motherwell, McLeish revealed a three-year deal is on the table for Jackson, who spent last season on loan at National League side Solihull Moors.
The Buddies boss said: “Louis is a player that we really like and we’re in discussions with him.
“We’re probably at the final stages of working out things because he is still contracted at Manchester United, he is working how that looks to end his contract with Manchester United and then finalise one with ourselves.
“Not quite there yet, so don’t know how that looks for this weekend.
“Hopefully I’ll get an update today but he’s one that we are in discussions with and we would like to add.
“He’s a Scotland Under-19 international, he should be in and around the Scotland Under-21 conversations as well.
“It follows our model of bringing in a younger-profile player and young Scottish players that we can develop for the national team as well.
“And it’s a long-term option for us at centre-back and one that if we can get it done, we’d be delighted with, but it’s not quite there yet.
“It would be permanent move. I think it’s a three-year contract that’s on the table for him which is the reason why we would do it.
“We are hoping to (add further) but it might depend on people going out to make room for that.”
The Buddies also secured the loan signing of former Aberdeen defender Calvin Ramsay from Liverpool on Tuesday.
McLeish said: “Yeah, delighted. He’s a player that’s normally well out of our reach.
“It’s a matter of circumstance that we can get him here.
“He’s coming into the last year of his contract at Liverpool, he needs a place to play and a platform to showcase what got him into Liverpool in the first place.
“We see already within the first few days of him training with us the kind of level he’s at, so we’re really looking forward to getting him involved in match days as well.”