The 25-year-old German international has attracted interest from across the continent, with United, Chelsea and Real Madrid all monitoring his situation at Stuttgart.
Juventus, coached by Luciano Spalletti, have now taken their pursuit a step further, holding talks with Stiller's representatives as they look to reinforce their midfield ahead of next season.
Tuttosport reports that Juventus plan to make a concrete move before the end of May, motivated in part by the existence of a release clause in Stiller's contract worth approximately £32 million — and crucially, the knowledge that VfB Stuttgart can cancel that clause by paying the player around £1.5 million directly, which would give the German club full control over any sale price.
Stuttgart are open to selling but are understood to want significantly more than the release clause figure, with reports suggesting they will hold out for closer to £50 million.
United view Stiller as a key part of a planned midfield overhaul following the confirmed departures of Casemiro and Uruguayan midfielder Manuel Ugarte this summer.
The German's profile — a deep-lying playmaker with elite passing accuracy and the ability to receive and distribute under pressure — fits what interim manager Michael Carrick's rebuild requires in the engine room.
Stiller opted against a move last summer in order to focus on his preparations for the World Cup, where Germany will be looking to progress beyond the group stage for the first time since winning the tournament in 2014. A strong tournament could further complicate any deal by driving his market value higher.