The sport's world governing body has never imposed such a sanction for an offence of discrimination, but Blatter - who tweeted his views on the matter on Sunday - insisted a fine was "not enough".
Blatter told FIFA's official website: "It is now up to us to take the adequate steps. What I feel we should do is give instructions to our national associations and the confederations - specifically to the disciplinary committees - to be very strong."
He added: "It is not enough to give a fine. Playing a game without spectators is one of the possible sanctions, but the best would be the deduction of points and the relegation of a team, because finally the club is responsible for their spectators."
Earlier this month FIFA imposed a one-match spectator ban on Bulgaria for racist abuse by their fans of a Denmark player. It was the first time the body had imposed anything more than a fine for racism.
Blatter has backed AC Milan midfielder Kevin-Prince Boateng for walking off the field during a friendly in response to continuing racist abuse, having initially appeared to question the player's decision.
He said: "It is a phenomena where football is a victim of our society. Discrimination and racism is everywhere in our society. We in football cannot be made responsible for what happens in our society.
"But nowhere in the world - regarding all the problems you can have in your private life, in business, in politics - can you solve a problem by running away. I agree with and support the movement of Boateng - as I have said - as it was a strong warning."