The 20-year-old was earmarked as a future replacement for Alexis Sanchez following his impressive performances in Ligue 1 this season but the Gunners have now backed away.
Goal understands Arsene Wenger's side are not willing to pay the French club's £44 million asking price for the Brazilian but do remain confident they could strike a deal in the summer.
Malcom is unsettled with his future up in the air and is keen to play under Wenger, but with Bordeaux, who find themselves a disappointing 13th in the Ligue 1 table, wanting to keep one of their star names it would seem a January transfer is unworkable.
Tottenham, meanwhile, have also emerged as a potential option for the player, with Malcom's representatives open to the possibility of discussions over a switch to Arsenal's north London rivals.
The Gunners have prioritised the signings of Henrikh Mkhitaryan and Pierre-Emerick Aubameyang, with both players having now agreed personal terms, and believe that the experience the duo possess can push them further up the Premier League table in the chase for a top-four spot.
Malcom has scored seven goals and laid on five more for his team-mates in the French top-flight in 2017-18, with Manchester United, as well as Spurs, having shown an interest in bringing him to the Premier League.
Speaking at his pre-match press conference on Thursday, Wenger revealed that he didn’t believe a January move for the Brazilian youngster was likely.
"I don't think we are close to [signing Malcom]," said Wenger. "I don't think [that it will happen in January], no."
Bordeaux president Stephane Martin, meanwhile, has said there was no formal contact between Malcom and Arsenal, with much of the early talks being conducted between the player's representatives and Arsenal officials.