The Spanish champions took the lead through Kylian Mbappe, but Bellingham was shown a red card for dissent towards the end of the first half.
Ante Budimir equalized for Osasuna with a penalty in the 58th minute, as Carlo Ancelotti's side endured their third consecutive league match without a win.
Atletico, who also played with 10 men after Pablo Barrios was sent off in the seventh minute, failed to take advantage of the situation, staying one point behind Los Blancos. However, Barcelona could draw level with the leaders if they defeat Rayo Vallecano on Monday.
Bellingham defended his actions, insisting he wasn't insulting the referee.
"I think you can see clearly in the video, I remember the incident very well, it was an expression to myself," he told the media.
"I'm not even directing myself towards the referee, but obviously, there was a misunderstanding. He believed I said something insulting to him.
"There was no intent to insult him, there was no insult."
Despite the draw, Real boss Carlo Ancelotti expressed satisfaction with his team's performance.
"In a footballing sense, we started very well, and we finished well despite having one fewer player," said Ancelotti.
The Italian coach made several rotations in preparation for the midweek Champions League second-leg clash against Manchester City, with Rodrygo resting and Brahim Diaz receiving a rare start.
Madrid dominated from the beginning, passing the ball swiftly, leaving Osasuna unable to make an impact.
It didn't take long for Madrid to break the deadlock, with Mbappe scoring his seventh goal in the last five La Liga matches after receiving a cross from Fede Valverde.
The game shifted when Bellingham was sent off in the 39th minute. The England international had words with referee Jose Munuera Montero, who then showed him a red card.
Ancelotti felt Bellingham should not have been dismissed.
"I think with Bellingham's red card, the referee didn't understand the English well. I don't think it was something offensive," Ancelotti told reporters.
"I think the red card came out because the referee was nervous. Bellingham did nothing today to deserve being sent off, absolutely nothing."
Real Madrid has voiced strong dissatisfaction with Spanish refereeing in recent weeks, calling it "rigged" and their frustration with refere Munuera Montero only grew in the second half.
Osasuna leveled the score before the hour mark with Budimir's penalty after Eduardo Camavinga fouled the Croatian striker.
Budimir's goal was his 13th of the season, placing him third in the league's scoring charts, behind Barcelona's Robert Lewandowski and Mbappe, who have 17.
Mbappe had a final opportunity in stoppage time, but Osasuna's goalkeeper, Sergio Herrera, denied him with a strong save.
Having played nearly an hour with 10 men, Real Madrid will hope the fatigue impacts them in Wednesday's crucial Champions League meeting with Man City at the Santiago Bernabeu.
Vincent Kompany has insisted Jamal Musiala is firmly on the right path despite not yet reaching his peak, after the midfielder delivered a standout performance in Bayern Munich's recent dominant 5-0 win over St. Pauli.
Manuel Neuer is set for another defining night as Bayern Munich prepare to host Real Madrid in Wednesday's UEFA Champions League quarter-final second leg at the Allianz Arena.
Real Madrid are facing a growing dilemma over Eduardo Camavinga, with the midfielder's future now in serious doubt at a crucial point in the season.
Harry Kane and Serge Gnabry handed Bayern Munich a timely lift after returning to training ahead of their decisive second-leg clash with Real Madrid.
