Lampard, who was celebrating with team mates after Chelsea's quarter-final win, ducked as the fan threw a punch. The spectator was manhandled to the ground before being led away by security men.
"These scenes were totally unacceptable and we will be contacting Spurs as a matter of priority," said an FA spokesman on Tuesday.
"We want to establish what safety and security provisions were in place for the game and to confirm what action was taken against the fans who ran onto the pitch.
"We understand that both individuals were arrested and we would expect them to be given football banning orders."
Chelsea will play Premier League rivals Blackburn Rovers in the semi-finals.
