Udogie was fouled by Diogo Jota in the incident that led to the Liverpool forward being sent off.
The 20-year-old Italian was then targeted with racist abuse on social media.
Tottenham said in a statement: “We are disgusted at the racist messages directed towards Destiny Udogie on social media following Saturday’s game against Liverpool.
“We will work with the Premier League and, where possible, take action against any individual we are able to identify.
“We stand with you, Destiny.”
A statement from Liverpool on Monday evening added: “Liverpool FC condemns all forms of racism and discrimination and will work with the relevant authorities and social media platforms to help identify those responsible and apply the strongest possible consequences.”
Tottenham beat Liverpool 2-1 after an injury-time own goal from Joel Matip.
Liverpool have since questioned the “sporting integrity” of the result after referees’ body PGMOL acknowledged a “significant human error” led to Luis Diaz’s goal being incorrectly disallowed for offside at 0-0.
Liverpool also had Curtis Jones sent off for a foul on Spurs midfielder Yves Bissouma.