The incident took place during the first half of England's 25-21 victory.
The Rugby Football Union said prop Marler, 25, apologised to Lee at half-time and was later "reminded of his responsibilities as an England player" by head coach Eddie Jones.
Six Nations officials are aware of the incident and establishing the facts.
They are yet to cite Marler but former England number eight and current Worcester director of rugby Dean Ryan said: "They'll have to act."
Speaking on BBC Wales' Scrum V programme, he added: "You can't brush it under the carpet. By apologising he is accepting that it's him, there is no denial, so you have to deal with what was said and the context of what was said, and it will be interesting to see how it moves forward."
The Welsh Rugby Union said in a statement: "We're disappointed by the comment made, and pleased Joe Marler apologised to Samson and was reminded of his responsibilities as an international rugby player."
The incident occurred after referee Craig Joubert gave a penalty against Wales for obstruction and a minor scuffle broke out.
Lee was surrounded by white shirts before Marler strode over with the words "Gypsy boy" heard clearly on the audio. The insult infuriated the Welshman, who has spoken about his heritage in the past.
Verbal abuse of a player based on religion, race, colour, national or ethnic origin, sexual orientation or otherwise carries a minimum sanction of a four-week suspension.
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England's Joe Marler apologised to Wales prop Samson Lee for calling him a "Gypsy boy" during Saturday's Six Nations match at Twickenham.