The Premier League rivals have been tipped with potential moves for the 20-year-old, who has attracted plenty of interest following his impressive performances in Ligue 1.
However, both the Gunners and Spurs are expected to fail in their attempts to sign the forward, with Poyet stopping the reports in their tracks.
"I asked straight away if Malcom would be staying or not," said Poyet, who was appointed Bordeaux head coach on Saturday.
"They said yes. I said: 'OK, now we can continue the discussion.'"
Bordeaux club president Stephane Martin added: "It has been said and said again: He will not be leaving.
"He has said he wants to play in the Premier League one day. That is normal. What player wouldn't want to play in England?
"There is no problem with Malcom."