Hector tweetedÂ after her side had beaten United 2-1 at the Olympic Legacy ParkÂ that she had been subjected to racist abuse during the match.
A statement on the FA Spokesperson Twitter account read: "The FA can confirm that we are aware of an allegation of discriminatory abuse during The FA Women's Championship match betweenÂ Sheffield UnitedÂ Women and Tottenham Hotspur Ladies on Sunday 6 January.
"We take all allegations of discrimination very seriously and will be making enquiries into the matter."
Tottenham confirmed they had reported the incident to the FA, while United said they had launched their own enquiry into the alleged abuse.
Hector, 23, wrote on Twitter: "Such a shame that racism seems to be rising up again in football - I received some monkey noises (on Sunday) from an opposition player. The only reaction was to let the football do the talking and that we did great start to the year! @ThlfcOfficial #COYSL #SHEvTOT #KickItOut".
The incident was reported by Hector to the referee during the game, Tottenham said.
A statement on their Twitter feed read: "The club is aware of allegations of racial abuse made by Renee Hector following (Sunday's) victory atÂ Sheffield UnitedÂ Women, that were reported to the referee by the player during the match.
"We have since reported the alleged incident to the FA and await their response. We shall be making no further comment on the allegations at this time."
Sheffield United added on Twitter: "Sheffield United are aware of the allegations made on social media following (Sunday's) fixture against Tottenham Hotspur Ladies. We can advise that the club is conducting a thorough investigation into this matter."
Tottenham's win lifted them to within a point of leaders Manchester United in the second tier.
The Blades are currently in seventh place in the Championship, with four wins from 11 matches this season.