Udogie was at the heart of two incidents that saw Diogo Jota receive two yellow cards, leading to a sending off for the Portuguese forward.
Tottenham maintained their unbeaten start to the season with a late win over the nine-man Reds after Joel Matip bundled into his own net late on.
Tottenham said on their official website: “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.”
Troy Townsend, head of player engagement at Anti-discrimination charity, Kick It Out, said: "No amount of controversy or high emotion surrounding a game can ever warrant this abhorrent outburst.
"We are extremely concerned that players are being continually targeted on social media following games in the worst possible terms and it is vital that those responsible are held to account. High-profile players have been targeted almost every weekend since the season began in August.
"This worrying trend is a sad reminder of how much work remains to combat discrimination in football, and it is vital that social media companies help address this problem with the seriousness it requires.
"We have received several reports regarding the racist abuse targeting Udogie and have shared these reports with the relevant authorities."
