FIFA secretary general Jerome Valcke said that it was "impossible" to change the structure of the tournament and that doing so would open FIFA to a host of other complaints of teams that felt they were cheated during qualifying.
"I think we have to be clear - Ireland will not be invited to play in the World Cup," Valcke said. "I would not say it is a nonsense but I mean it's impossible.
"There is no hope to give, or to have, to think that there will be more than 32 teams playing in the World Cup in 2010 in South Africa."
FIFA president Sepp Blatter said Monday that the Irish had asked football's world governing body to be let into the 2010 tournament because of Henry's handball.
Valcke said Irish Football Association officials met with Blatter at FIFA headquarters in Zurich several days ago.
"The FIFA president was very clear by saying it was impossible, because if we do so, why not Costa Rica and others?" Valcke said at the start of a week of meetings centered on Friday's draw for the final tournament.
Costa Rica also complained to FIFA that a goal which helped Uruguay win a playoff between the two nations was offside.
Valcke said Blatter promised to put the official Irish request before FIFA's executive committee members who meet in an urgent session Wednesday.
The 24-member ruling panel is set to discuss issues arising out of the playoff matches in Europe, Africa and Latin America last month - including Henry's handball.
It could decide to give football's rules making body, which meets in Zurich next March, the authority to speed up trials with five match officials using an additional linesman behind each goal to help spot infringements in the penalty area.