The 22-year-old centre-back has emerged as an appealing, cost-effective option for several clubs seeking defensive depth, though Leicester are holding out for a £10 million fee to sanction his departure from the King Power Stadium.
Ahead of their return to the Champions League scene, Manchester United are undergoing a squad rebuild under Michael Carrick.
With several fringe players expected to depart, United view Nelson as a high-upside backup who can develop within their system.
Meanwhile, West Ham are eager to strengthen their central defence, identifying the 1.95m defender as an ideal target.
Leicester City are under some pressure to sell following their relegation to League One, and Nelson is under contract until June 2027.
However, the Foxes will only agree to a deal on their terms after Italian side Torino made a formal opening bid of around €7 million (£6m) for the England youth international, which was promptly rejected by the Leicester hierarchy.
For Nelson, this top-flight interest caps a steady rise through the developmental ranks. After successful loan spells at Oxford United and Doncaster Rovers, the composed defender made 27 competitive appearances for Leicester last season.
While Nelson was previously open to a move to Serie A under Ignazio Abate at Torino, interest from Manchester United and West Ham has shifted the transfer dynamic, with neither English club expected to be deterred by his £10 million valuation.