The 21-year-old was set to join the Italians this week but Spanish champions Barca's 11th hour offer changed Malcom's mind and he became their third signing of the transfer window.
Malcom, who has yet to represent his country at senior level, began his career in his homeland with Corinthians and was part of the Campeonato title-winning side of 2015 before moving to France in January 2016.
"I informed [Martin] and [Malcom's] agents that Roma were ready to beat Barcelona's offer".
# Bordeaux booked a flight to Roma for Malcom to travel for his medical.
"I told him that we had an agreement, we had made the deal, but he said that nothing had been signed and that, despite having exchanged some documents to be signed, nothing could be done".
# Roma then sent an improved bid. "I can't wait to get started". If someone wants to come to Roma that is great, but ultimately if they do not want to come to Rome then we do not want them.
"After that meeting, I was surprised to be told that the agents and Bordeaux wanted another improved offer, as Barcelona were still pushing for a deal", Monchi said.
In an interview on Roma's website, Sporting Director Monchi said that Roma are considering legal action after a proposed deal collapsed.
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"Within the club, we are looking at our options and seeing if we have a legal case", Monchi told Roma TV.
"President [James] Pallotta made the decision to make the best offer possible, but when the negotiation became an auction we chose to withdraw from it".
Monchi does not seem a happy man and given the fact that he mentioned the messages he received from Malcom's agent and Bordeaux's president, it seems like he is willing to take legal recourse.
"This is a boyhood dream come true", said Malcom, who is yet to be capped by Brazil.
Following his medical Malcom is expected to join up with his new team-mates on their pre-season tour in the United States.
Malcom scored 12 goals and registered eight assists in 38 appearances for Bordeaux in Ligue 1 last season - the French club thanking him for his "hard work and commitment" and wishing him "all the best for this new adventure".
Malcom joins compatriot Arthur - also uncapped - in the Barca squad for the 2018-19 campaign, with the Catalans getting their title defence underway against Deportivo Alaves on August 19.
They have also signed Sevilla centre-back Clement Lenglet for 35.9 million euros (£32m).