Blues co-managers Johnson and Bernard Morley held 'informal' talks with the highly-rated 25-year-old Stockport County striker, who is seeking pastures new after netting 25 goals for the Hatters last season.
The Blues had hoped that former Newtown striker Oswell, who works as a physiotherapist at the Countess of Chester Hospital, would look to remain part-time and combine his career in the NHS with his football.
But after talks with Oswell, Johnson believes that the striker has his heart set on taking a shot at the full-time game, with Port Vale, Morecambe, Wrexham and Hartlepool United all keen.
Said Johnson: "We love him as a centre forward and thought he was superb last season, not just in terms of the goals he scored but in terms of how he played against us (when Salford City managers).
"We met with Jason for a chat last week, very informal, just to see what he is doing as he is a free agent. No money was spoken about but he made it pretty clear he wanted full time football. The stumbling block is that he has a decent career at the Countess in Chester but he does want to give it a go and he does believe in his ability, and I believe in his ability.
"It is looking like he will probably go down that route if I'm honest."
Johnson and Morley are aiming to add some more bodies to their squad over the coming days ahead of the players reporting back for pre-season training next week.
"We are back in on Monday night and we would like to have 75% of what the squad will be that starts the season in training with us on Monday," said Johnson.
"There will be a handful of trialists there as well that we can run the rule over, but if we get the best part of four or five players this week that are on that list then we will be delighted."