Everton's bid for Malcom currently comes close to Bordeaux's asking price, while Roma's offer may need to be adjusted to compete with the Toffees.
Roma are launching a push to make sure they beat Everton to Malcom's signature and held talks with the winger's agents on Thursday night.
An agreement was found on the basis of a five-year contract worth €3m per season, with Malcom happy to make the move.
Now Roma are approaching Bordeaux and have proposed a €35m deal, including bonuses, and are expecting a reponse in the next 24 to 48 hours.
According to Sky Italia, Everton's offer is closer to Bordeaux's asking price of €40m.
Roma may therefore have to work on their proposal to make it more attractive for Bordeaux.
The Giallorossi are boosted by the fact that Malcom wants to move to the Italian capital and prefers Serie A over the Premier League.
But if Roma cannot agree a fee with Bordeaux, Malcom may find Everton are the only show left in town.