"I always said I wanted Argentina to get to the final because Brazil would be there but it never worked out like that," Neymar told reporters on Thursday.
"I still want them there because my two team mates are there, Messi and Mascherano, and I hope they win."
Rodriguez, however, railed at Neymar's support.
"I'm indifferent to Neymar saying that Argentina or Germany should be champions," Rodriguez said. "What bothers me is what they say there (in Argentina), I don't care what outsiders say."
The Brazil v Argentina rivalry is one of the hottest in football and is particularly tense this week as Argentina are within sight of winning their third World Cup at Rio's famous Maracana stadium.
Rodriguez, who is expected to play at least some part in the final against Germany, made his less than magnanimous comments just hours after Neymar spoke.
Argentina will play Germany, who handed Brazil their heaviest ever World Cup defeat in Tuesday's 7-1 semi-final thrashing, at the Maracana on Sunday.
Brazil take on the Netherlands in the third-place playoff on Saturday but Neymar will miss the game, as he did the semi-final, because of a back injury.
Neymar singled out Messi for special praise and said the World Cup trophy would be a fitting addition to his bulging trophy cabinet.
"He's won almost everything and I think he deserves to be champion and I am cheering for him to be champion," Neymar said, jokingly adding that he was supporting his friends rather than Brazil's biggest rivals. "He is my team mate and my friend."
Brazil have won the World Cup five times and Argentina two.
(Writing by Andrew Downie, additional reporting by Luis Ampuero in Belo Horizonte, editing by Ed Osmond)
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