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Madrid opened the game dominating possession, but were unable to create many real chances inside the opening 10 minutes.
Alvaro Cejudo whipped in a dangerous ball on 20 minutes but the Real defence managed to clear after a scramble in the box.
The home side managed a spell of possession towards the end of the half, but a lack of movement in the final third meant they were unable to create any chances until the whistle.
Osasuna were giving as good as they were getting with passing in the middle and created a good opening when Emiliano Armenteros's low cross was missed by everyone in the box.
They then forced a series of corners but the Real defence was not allowing any way through.
With 15 minutes left Enrique Sola looked to have the chance to break the deadlock but Alonso got back to make a vital block.
Then substitute Kaka was sent off for two bookable offences having been on the pitch for just 15 minutes.
With 10 minutes left Jose Callejon had the ball in the net, but his low finish was ruled out for a marginal offside.
Despite a late penalty shout for the home side, there was to be no goal and the points were shared.