Gunners boss Unai Emery approached United and offered to take the defender on loan for the rest of the season.
But United rejected the approach out of hand as they do not want to strengthen a top-four rival club.
Arsenal are short of numbers at the back after Hector Bellerin was ruled out for the season yesterday.
The Spaniard ruptured his anterior cruciate ligament and is expected to be out for between six and nine months.
He joins Rob Holding who was also ruled out for the remainder of the campaign before Christmas.
Emery viewed Bailly as a viable replacement who would have joined Sokratis, Shkodran Mustafi and Laurent Koscielny.
The Ivory Coast international has made just six Premier League starts so far this season.
Ole Gunnar Solskjaer gave Bailly a rare start against Bournemouth but he was sent off for a dangerous tackle.
The £30m signing has less than 18 months left on his contract and he will stay to fight for his place.
Emery is desperate to strengthen his Arsenal squad but is hampered by a lack of transfer funds.
Meanwhile, Arsenal’s head of recruitment Sven Mislintat will leave the club in February after just 14 months.
The German, 46, joined the Gunners in December 2017 from Borussia Dortmund and was involved in all eight of the signings Arsenal have made since then.
In a statement, Arsenal said Mislintat had done a “truly outstanding job”.
“It’s been an amazing experience to work at a great club like Arsenal in what has been a big time of change,” said Mislintat, who will leave on 8 February.
“I am excited about what the future holds for the club and am looking forward to new challenges for myself.”