The Gunners were intent on trying to bring Hummels to the Emirates following his heroics at the World Cup for Germany.
But the bumper fee on his head - believed to be in excess of £30m - has priced them out of a deal.
Arsene Wenger has instead moved for Villarreal's 24-year-old centre-back Gabriel Paulista, who is due to complete his move today.
Arsenal will pay around £11m for the Brazilian's services.
Paulista is set to wear the No.5 shirt previously donned by Steve Bould, Martin Keown and Kolo Toure.
The deal means United are now able to mount a bid for Hummels without fear of it being hijacked by Arsenal.
Louis van Gaal is a huge fan of the 26-year-old and is desperate to stage a swoop to shore up his defence.
Hummels reacted angrily when asked about whether he would leave at the end of the season by German newspaper Bild.
He said: "Are you looking for a headline?
"No, I will keep it the way I've been doing it for seven years.
"I'll let everything come to me and just look forward only to the start against Leverkusen [on Saturday] and the second half of the season."