The Italian outlet reports that the Red Devils are in first place in the race to sign the 29-year-old Moroccan, who has fallen down the pecking order at Stamford Bridge.
Ziyech has made just nine appearances across competitions this season, totaling a mere 270 minutes of action.
AC Milan are interested in the former Ajax attacker, but a reunion with Erik ten Hag at Manchester United seems the most likely destination.
Ziyech's departure from west London is anticipated for the January transfer window.
The forward is eager to leave Chelsea for more playing time - he produced the best football of his career under Ten Hag at Ajax.
He made 100 appearances under the Dutch coach, scoring 35 goals and creating 53 assists.
Ziyech can play on both flanks and in an attacking midfield role - his versatility would be of great benefit to the Red Devils.
However, it remains to be seen whether the Blues would be willing to sell the Moroccan to one of their top four rivals.
Manchester United are currently fifth in the league, five points ahead of eighth-placed Chelsea.
Ziyech's current deal at Stamford Bridge expires in 2025, and he is on wages of £100,000-per-week, according to the Sun.
Manchester United defender Raphael Varane explains his reaction when picking up an injury against Chelsea
Varane was forced off with a leg injury during Manchester United's 1-1 draw with Chelsea on 22 October.
The French defender was distraught when leaving the pitch as the fear of missing the 2022 FIFA World Cup crept in.
Luckily, the United center-back has featured during France's World Cup campaign in Qatar, making two appearances for Les Bleus.
Didier Deschamps' side have qualified for the last 16 and will face Poland.
However, Varane has explained why he was so tearful in reaction to picking up the knock in the draw with Chelsea.
He said (via Manchester Evening News):
“It was a difficult moment. At first I thought I had a more serious injury than it really was. I thought it was over for the World Cup. A sportsman’s life has ups and downs, you have to pick yourself up and come back stronger."
Varane then discussed his recovery, saying:
“I went into fast-track recovery mode, I put in a lot of effort and made a lot of sacrifices to be ready in time. It was difficult but I’m making the most of every moment. I know exactly what it means to wear the France jersey. I was desperate to overcome [the injury].”