The teenage attacker's agent confirmed this week that personal terms had been struck with Real Madrid, though conceded nothing had yet been agreed between the two clubs.
TMW says the revelation has kicked both Manchester clubs into action and talks have been revived with Mallorca over the last 24 hours over a fee for Asensio.
United are willing to spend big to sign Asensio in January and take him immediately to Old Trafford. City, in contrast, would prefer to wait until the summer before finalising a deal.
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