The Real Madrid star won the Champions League in his first season at the Bernabeu but with United willing to spend heavily again in the transfer market, he could be tempted back to English football.
They face a battle to convince Bale, however, as the player has settled well in Spain.
However, the deal won't be done until the summer as captain Hummels wants to see out the season and keep Dortmund's Champions League dreams alive.
If their Hummels pursuit failss, United will turn their attention to Valencia's Nicolas Otamendi, who is available for £40m.
The deal relies on Valencia securing Marseille centre-half Doria as a replacement, so this transfer may only reach its conclusion at the end of January.
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Gareth Bale could be heading to Manchester in the summer
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