The 30-year-old centre-back revealed United's apparent interest in an interview with Fox Sports Brazil, although his agent Marcus Marin has moved to play down talk of the Brazil international moving to Old Trafford.
The Guardian quotes Dante as telling the TV station in his homeland: "I'm thankful for the Manchester United interest. It will all be sorted out by the end of the month.
"We have to wait until 31 January because there is a lot going on."
Despite his client's comments, Dante's agent, Marin, is focused on securing a new deal with Bayern.
"There is no agreement with Manchester United. He is not going to leave Bayern in the window," said Marin.
"At the moment Bayern are the only people we have spoken to anyway because we are talking over a new contract.
"We are still waiting for an answer from them and, although things happen very quickly in football, until that situation changes, we cannot speak to anyone else anyway.
"We are hoping that something can be resolved before the World Cup."
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