The Old Trafford club have been heavily linked with the defender since before the World Cup, and Louis van Gaal remains keen to bring him to the club at the earliest opportunity.
But Arsenal, as reported by the Daily Express, are looking for reinforcements of their own in January, and are ready to steal a march on their Premier League rivals with an opening £32m offer.
Hummels turned down the chance to leave Dortmund over the summer, when United and Arsenal were interested. But van Gaal wants the Germany international to help push his side into the top four and Champions League qualification for next season.
Van Gaal has been short in defence following injuries to key players - Chris Smalling, Jonny Evans and Phil Jones have all spent time on the treatment table - forcing the United boss to call up Tyler Blackett and Paddy McNair
The report claims that could be a key factor in Hummels' decision, as the defender will not move to a team that is not in Europe's elite club competition, or not in with a real chance of qualifying for next season.
Arsenal in the transfer market
Wenger has something of a defensive crisis on his hands, as the departure of Thomas Vermaelen has left him with Per Mertesacker and Laurent Koscielny as the club's only two senior centre-backs.
Dortmund are desperate to keep their squad together having lost Mario Gotze and Robert Lewandowski in consecutive summers.
Such a thin back-line was always liable to come under pressure should injuries hit, and the likes of Nacho Monreal and Calum Chambers have been forced to step in at various points this season.
The Arsenal boss would have to spend big to land the Dortmund star though, as the German giants are desperate to keep their squad together having lost Mario Gotze and Robert Lewandowski in consecutive summers.
United in the transfer market
As for United, van Gaal has been short in defence following injuries to key players - Chris Smalling, Jonny Evans and Phil Jones have all spent time on the treatment table - forcing the United boss to call up Tyler Blackett and Paddy McNair. McNair is now also injured.
The departures of Nemanja Vidic and Rio Ferdinand over the summer opened up an opportunity for United to add an experienced defender to their squad, but they chose instead to sign Marcos Rojo from Sporting Lisbon.