Real begin their quest to end a poor run in Europe's elite club competition at home to Ajax Amsterdam on Wednesday.
Mourinho, who won the Champions League with Porto in 2004 and Inter Milan last season, wants to be the first coach to claim the European Cup with three different clubs.
Real have failed to progress beyond the last 16 in the past six seasons and club President Florentino Perez has spent around 350 million euros on players and to lure Mourinho to the Bernabeu in a bid to restore success in Europe.
"Real Madrid want their 10th just as much as I want my third but we both know it's very difficult," the Portuguese former Chelsea manager told a news conference.
"There's a group of six, seven or eight teams that have what it takes to win this title," he said.
"Even though it's our first season together and we are in a period of construction, if we get past the opening round and get to the last 16 we'll start to get stronger and be more positive about the future."
Mourinho and his players are under pressure to entertain their fans on Wednesday after Saturday's laboured 1-0 La Liga win at home to Osasuna was greeted by impatient whistling from some supporters.
Ajax, four-times champions of Europe but making their first Champions League appearance since 2005/06, will not be easy opponents, Mourinho cautioned.
"Ajax is a great club with a great tradition," he said. "Both they and Real Madrid are monsters in the history of this competition. I am a big fan of the youth school (Ajax) has and they will be a difficult team to play."
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