The Lilywhites were dealt a big blow in their aim to become top dogs in North London for a second season running when falling to a 2-0 loss at the Emirates Stadium on Saturday.
Pochettino still stands by his assertion that success in the top flight and Champions League counts for much more, however, telling The Times: "You arrive at a big club and you can play a final and win - that happens at Liverpool, Arsenal, Manchester United, Manchester City, Chelsea.
"At all big clubs they try to use the whole squad for the cups, but you know the Premier League and Champions League is a massive challenge for big clubs - this type of trophy are the real trophies. Would I swap for the FA Cup wins? No, because Tottenham is Tottenham and Arsenal are in another stage of their project.
"I don't care where they are, because after three and a half years we have achieved a lot. I would like to be first and not second and win cups - but we are in a different process of our project. After three and a half years we have a team that can compete with Real Madrid.
"That is pressure we are building, to win only cups is not useful for us. If we can win the FA Cup, Carabao Cup fantastic. But the principal option is to win the Premier League and put Tottenham in a position to win the Champions League."
Spurs remain one point above Arsenal in fourth place following this weekend's loss.