The YEP understands the Whites offered a substantial loan fee and £20m in the summer, if promotion to the Premier League was secured.
Saints boss Ralph Hasenhuttl has reiterated a number of times that Adams is an important part of his strikeforce at St Mary's and will be going nowhere and the latest offer from Elland Road has been knocked back.
Adams is yet to score in the Premier League but fired in 22 goals for Birmingham City in the Championship last season and has been viewed as an ideal addition to Marcelo Bielsa's squad.
Leeds remain in talks with clubs both domestically and on the continent, working on a number of deals, and are confident they can land a striker before next Friday's transfer deadline.
A loan-to-buy deal appears to be the likeliest route to a new signing, given the restrictions on spending brought about by Profit and Sustainability [P&S] rules and Leeds' determination to avoid a breach that would risk a sporting sanction, like the nine-point deduction handed out to Birmingham last season.
It would take a transfer fee well in excess of the current Championship record of £15.8m - paid by Wolves in 2017 for Ruben Neves from FC Porto - to buy the kind of player Leeds are looking at, outright. And that sum would take most, if not all of the division's clubs over the P&S limits.
A number of Premier League strikers have been linked to the club, but the likes of Adams and Watford's Andre Gray are not currently for sale, in what has been a fairly stagnant January market so far. There may yet be movement, however, that allows Championship clubs to get their targets in the latter stage of the window.
Meanwhile, United are hopeful of concluding a deal for Manchester City winger Ian Poveda later this week.
Yesterday they confirmed the arrival of 18-year-old goalkeeper Elia Caprile from Chievo Verona.