Scotland Yard said that after assessing the messages, that referred to the Holocaust, an inquiry had been launched as part of Operation Arrowtip, which deals with football-related crime.
The force is also liaising with both clubs over objects that were thrown.
"Having assessed messages published on Twitter, officers from Operation Arrowtip are now investigating."
Walcott was apparently pelted with coins as he was stretchered off after injuring his knee during the north London derby.
The force said: "The Metropolitan Police Service (MPS) is aware of incidents during the Arsenal Vs Tottenham Hotspur match on Saturday, January 4 where a number of objects were thrown from both home and away sections of the crowd.
"The MPS takes all such incidents very seriously. Officers from Operation Arrowtip are investigating and are liaising with both clubs and the Football Association (FA) over these incidents."
The FA confirmed that it has contacted Arsenal and Spurs over the missile-throwing and said it will back the harshest punishments possible, including life bans.