Brentford have rejected two offers for Mbeumo, the second worth an initial £55million plus £7.5m in add-ons, as they hold out for a higher fee than the £62.5m that United paid for Matheus Cunha.
Mbeumo has already reported for pre-season training at Brentford this week, which has sparked further concerns among supporters.
However, the report adds that his desire is to join United, even after a late approach from Tottenham Hotspur following the appointment of Thomas Frank as their new head coach.
It's claimed that United believed they were nearing a breakthrough in their negotiations for Mbeumo last month after the 25-year-old opted for a move to Old Trafford.
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However, the appointment of Keith Andrews as Frank's successor has caused a delay, with Brentford reluctant to lose their top players as well as their boss.
Mbeumo's contract with Brentford extends until 2026, with the club holding an option for an additional year. The Cameroon international switched agents last year with the intention of departing.
Following their first pre-season friendly against Leeds United in Stockholm on Saturday, United will head to Chicago on Tuesday, where they will spend most of their three-match tour.
United's inaugural match in the Premier League Summer Series will be against West Ham United in New Jersey on July 26, before returning to Chicago to face Bournemouth. Their tour concludes against Everton in Atlanta on August 3.
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If United fail to reach an agreement for Mbeumo before their departure to Chicago, the winger could potentially join them later during their 12-day tour.
Andre Onana joined the United squad in the United States for their 2023 pre-season tour, featuring in their second and third friendlies against Real Madrid and Borussia Dortmund.
But Ruben Amorim will want a second major summer signing in his squad by the time they travel across the Atlantic.