The 17-year-old burst onto the scene in Scottish League One after making his debut in men's first-team football as a 16-year-old schoolboy in July 2022.
One appeared 17 times for Hamilton in the 2022-23 term and has commenced this season impressively, scoring two goals in four matches in third tier of Scotland.
The youngster has represented his nation at youth level, having played five matches for Scotland Under-17s since earning his maiden cap in October 2022.
According to the Daily Mail, Sheffield United have taken One on trial in the closing stages of the month, with a view to him potentially joining the Bramall Lane club.
The report claims that the forward has attracted interest from elsewhere already this summer, with Brighton & Hove Albion said to be keen on the player.
It is understood that the Seagulls had a £150,000 bid for One rejected by Hamilton in July, giving the Blades an idea of how much they would need to spend in order to secure his services.
Nottingham Forest and Burnley are also believed to be keeping tabs of the youngster, with the Premier League duo looking to lure the Scottish talent to their respective ranks.
The report claims that Steve Cooper's Tricky Trees are also working on a move for fellow Hamilton prospect Cormac Daly - a 16-year-old winger who is highly rated in Scotland.
There are also admirers of One in the English Football League, as Championship side Blackburn Rovers are said to be willing to take the 17-year-old this summer.
Blades boss Paul Heckingbottom has openly expressed his desire to add more firepower to his side's attack before the transfer window slams shut on Friday night, and the arrival of Cameron Archer will have pleased supporters of the Yorkshire club.
The 21-year-old sealed a switch to Bramall Lane from Aston Villa for a fee in the region of £18m on Sunday morning, and the striker will now hope to replicate the goalscoring ability he showed whilst on loan at Middlesbrough last term.
One is said to be have started a three-day trial with United on Monday, although it is very unlikely that Heckingbottom views the Scotsman as someone who could play in the Premier League this season.