The report added he is "furious" after Tuchel told him to lose weight and he now wants out of Paris.
Verratti's agent is Mino Raiola, a number of whose clients have joined United in recent years, including Zlatan Ibrahimovic, Paul Pogba, Henrikh Mkhitaryan and Romelu Lukaku.
If the 25-year-old is looking for a way out of PSG, it would be no surprise if Old Trafford was a potential destination.
And United could be improved by the addition of Verratti.
He remains ill-disciplined, but the 25-year-old is a gifted technical midfielder.
A deep-lying playmaker, he could fill the role vacated by Michael Carrick following his retirement at the end of last season.
United have already added a new man to their central ranks, though, after agreeing to sign Brazilian midfielder Fred from Shakhtar Donetsk earlier in June.
That does not necessarily rule out the Premier League giants from signing Verratti, but manager Jose Mourinho will likely be eager to first address other areas, most pressingly the centre of defence.
Furthermore, the Italy international still has a contract with PSG until 2021, so even if the French champions are open to selling him, they may demand a fee that proves restrictive.