The England international has been heavily linked with Manchester City and is thought to be keen on a move to the Etihad Stadium.
However, City have yet to reach an agreement with Leicester, having so far failed to submit an acceptable bid.
That has allowed United to march to the front of the queue, with the Red Devils offering Leicester £70m.
The Foxes insist they do not need to sell and would not settle for less than the £75m Liverpool paid for Virgil van Dijk in January 2018.
United’s offer will test their resolve, though, as well as forcing Manchester City to respond.
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