Real suffered only their second loss of the season 24 hours after leaders Barcelona racked up their 15th consecutive win, slipping seven points behind their arch-rivals.
"The result is tough but not definitive," Real director general Jorge Valdano told reporters.
"At the midway point of the season it isn't possible for the championship to be over for Real Madrid. It is well known our capacity to fight and to never give up in these types of situations."
Real had a tough King's Cup semi-final first leg at Sevilla last Wednesday which they won 1-0 and the return is this week but they have lacked a cutting edge in recent matches.
They struggled to control Osasuna and missed Xabi Alonso's influence in midfield as he was kept on the bench after a bout of flu, only appearing when they were chasing the game in the second half.
"The situation is much more difficult with regards to Barcelona, but come Wednesday we'll only be thinking of that game," coach Jose Mourinho said.
"My players worked hard and I have no complaints. We lost to a rival who gave their all, who have had a week to prepare, and who pushed themselves to the very limit.
"I have to congratulate them. I'm not saying they deserved to win but they deserved a draw."
It was only Real's second defeat of the season after their 5-0 hammering at Barcelona in November.
"We have to think positive," top-scorer Cristiano Ronaldo said. "It's difficult but we have to think positive."
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