The Red Devils officially announced on Tuesday that they had found an agreement with Real Madrid to sign the 28-year-old centre-back.
Varane will now undergo a medical and agree personal terms before signing with the Premier League giants.
ESPN says the news has caused Bailly to re-consider his future with the club.
Bailly only signed a long-term extension with United in April, keeping him tied to the club until 2024.
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