Dusan Vlahovic celebrates with Juventus.
United have been linked with a swap deal involving Vlahovic and Zirkzee.
Manchester United return to action in Sunday's Manchester derby against Manchester City.
Ruben Amorim's side will be looking to avoid a third consecutive Premier League defeat when they head to the Etihad Stadium this weekend. United lost 2-0 at Arsenal before last Saturday's surprise 3-2 defeat to Nottingham Forest at Old Trafford.
The Reds will have been boosted by their 2-1 win at Viktoria Plzen in the Europa League last time out, but there's still much work for Amorim to do on a squad which sits 13th in the top-flight standings after 15 matches.
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One player who has been linked with making the switch to M16 is Juventus striker Dusan Vlahovic. A report from Italian journalist Graziano Campi reckons that a swap deal with Joshua Zirkzee is possible - despite the Dutchman only arriving in Manchester this summer.
"A Vlahovic-Zirkzee swap, in my opinion, is absolutely not science fiction, because Vlahovic has not yet renewed [his contract]," he told Calciomercato.it. "If he were to renew, let's remember that he is the Serie A striker who has scored the most goals in 2024."
Campi also claims that Victor Lindelof could be of interest to the Serie A giants. The Sweden captain has played only 58 minutes in the Premier League this season.
"There are many names in defence; in my opinion there could be Lindelof from United and watch out for the Red Devils who want to do some market operations with Juventus, but more in the summer," he added.
Elsewhere, Spanish publication Fichajes state that Real Madrid have decided not to pursue AC Milan player Theo Hernandez, leaving the path clear for United to pursue the £42million rated left-back. The report claims that Real are focussed on other priorities - and the report states that "clears the air for United" to strengthen their defence with his signature.
United are in need of a new left-back who can play at wing-back in Amorim's preferred 3-4-3 system. Luke Shaw has struggled with injuries, while Tyrell Malacia is coming back from a long-term absence.