La Volpe already has two naturalised players in his squad -- Brazilian-born midfielder Antonio Naelson and Argentine-born foward Guillermo Franco -- and the subject has split Mexican football down the middle.
Critics are especially angry because La Volpe has overlooked Mexican-born midfielder Cuauhtemoc Blanco and striker Adolfo Bautista who many feel deserve a place in the squad.
The volatile Blanco in particular is regarded by many as Mexico's best player.
On Friday Gaitan told the sports daily Record that he had met La Volpe and said he identified more with Mexico than the country of his birth.
'I've known for a long time that it's very unlikely (to be picked for Argentina) and I want to play football I want to dream about going to the World Cup' he said.
'I love Argentina but now I spend more time here and I identify more closely with the people of Mexico. If I have a chance to represent them I will do my best.'
Earlier this week La Volpe told the sports newspaper Record: 'The media are very enthusiastic about Cuauhtemoc and Bofo (Bautista) but I believe that if in the last two championships we have to talk about a goalscoring midfielder then it is Gaitan.
'As far as I'm concerned he's Mexican.'
Naelson who has lived in the country for nearly 10 years and is married to a Mexican was the first foreign-born player to be picked by La Volpe.
He was a regular in the qualifiers and became the first naturalised player to score for Mexico when he was on target in a 3-0 win over Trinidad & Tobago.
Franco who plays for Monterrey was then picked to face Guatemala in October when Mexico had already qualified. Unlike Naelson he has only been in the country for three years.
Former Mexico international Hugo Sanchez has led the crusade against the use of naturalised players saying he would be saddened if his country were to win the World Cup with a team of players born outside Mexico.
Toluca president Rafael Lebrija said: 'I don't agree because the only naturalised player who played in the team which qualified was Naelson. Since then he (La Volpe) has picked another and now he wants yet another.'
The media have also criticised La Volpe's policy.
Meanwhile Blanco seems further than ever from his third World Cup after La Volpe said that he had to work harder.
Blanco has not been picked since he declined to play at last year's Confederations Cup in Germany saying he needed a rest.
'I haven't got any sort of problem with La Volpe but if the coach doesn't want to pick me what can I do' said Blanco. 'I know I'm one of the best.'