Foderingham made 10 saves at the Tottenham Hotspur Stadium, only to see Spurs score in the 98th and 100th minutes deep into added time, to win 2-1.
After the result, the goalkeeper posted on Instagram: "I don't mind opposition fans calling me every name under the sun. But be easy with the racism and family threats. Think before you type."
The Blades responded on Sunday morning offering support for their keeper and promising to investigate.
A statement read: "Sheffield United condemn the racist, abusive and threatening messages that have been sent to Wes Foderingham after yesterday's game against Spurs.
"The club will now work with relevant bodies to investigate and support Wes.
"There is no room for racism in our game."
Spurs then added in a statement of their own: “We are disgusted to hear of racist, abusive and threatening messages sent to Wes Foderingham following yesterday’s match.
“The Club stands firmly against all forms of discrimination and will cooperate fully with Sheffield United and the relevant bodies in their investigations.
“We shall not hesitate in taking the strongest possible action, including Club bans, against any so-called fan found responsible.”