The Red Devils are long-term admirers of the Brazil international, having previously shown interest before he completed a move to the Allianz Arena from Borussia Monchengladbach in 2012, and continue to monitor his progress with the European champions.
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Goal UK understands that Bayern is receptive to offers for the 30-year-old, having so far been unable to reach an agreement on an extension to the defender's current contract, which expires at the end of next season.
After being asked by Fox Sport Brasil if he had already signed for United, Dante said: "Not yet. I am still a Bayern player. I appreciate Manchester United's interest. At the end of the month it will be resolved."
But while Dante suggests a move to Old Trafford is close, his agent was keen to stress that at the moment he has not held talks with any club but Bayern, although he did acknowledge that English soccer interests his client.
"He has two more years on his contract and we are now in the race to make a new contract, but at the moment we have not found a solution so we are still waiting for Bayern," Marcus Marin told Goal. "We are not in touch with any other clubs at the moment but in football you never know."
When asked specifically about Dante's quotes on Brazilian television and Manchester United's supposed interest, Marin replied: "No, I know that they follow him and we were in touch with Manchester [United] two years ago before [Dante] went to Bayern Munich, but at the moment I know only that they watch him at games, but we haven’t spoken to the club at the moment.
"We have not found a solution [with Bayern], so we are still in the race, but England is very interesting, Manchester United are a big club so you never know. Next month [Bayern] play in the next round of the Champions League against Arsenal and it is a good way to present himself in England. In the last game [in England] he played very well so yeah, we will see."Dante was part of the Bayern Munich team that won the Bundesliga, DFB-Pokal, Champions League, UEFA Super Cup and Club World Cup in 2013, as well as the Brazil side that claimed the Confederations Cup last summer.
United's search for reinforcements has taken manager David Moyes across Europe, with the Scot present at Cagliari versus Juventus earlier this month to run the rule over home defender Davide Astori - as well as Juve midfielder Claudio Marchisio.
The Premier League champions have also held talks with Benfica regarding Argentine defender Ezequiel Garay.
Veteran defenders Rio Ferdinand and Nemanja Vidic look set to leave Old Trafford this summer having not been offered extensions to their current deals, which expire at the end of the season.