The Bernabeu outfit have not lifted Europe's most prestigious club title since 2002, which also remains the last time they reached the final.
Arch rivals Barcelona have won the competition on three occasions since then, but Ramos insists the quest for continental glory will not become an "obsession" for the men from the Spanish capital.
"It's clearly a competition that’s very dear to the club, and it's been a long time now since we held that title," he told FIFA.com. "I'm not sure it's an obsession but the desire to win it is greater than for any other competition.
"I think it would be wrong though to let it become an obsession."
Real, who have won a record nine European Cup crowns, face Schalke over two legs in the round of 16 of this year's tournament, and Ramos knows the importance of taking each game as it comes.
"We need to just take it step by step, like we’ve done in recent years – reaching the semi-final and almost making the final," he added. "If we can go all the way, then even better, no?
"It’s tricky, but Real Madrid are obliged to fight till the death for that trophy."
Carlo Ancelotti's men currently sit one point behind Barcelona and Atletico Madrid at the top of La Liga.
