Conte has been linked with Julen Lopetegui's job, with the Real coach's future in serious doubt after yesterday's home defeat to Levante.
Marca was particularly hard-hitting, declaring "This Madrid are a wreck" after Lopetegui's team saw their run without a victory extend to five matches.
Marcelo ended the streak in the second half at Santiago Bernabeu Stadium, but the negative milestone had already been set, coming three months after Ronaldo left to join Juventus.
And despite some decent performances that followed, Real Madrid have gone without scoring in their last four games losing three in the process and one ending in a stalemate against Diego Simeone's Atletico Madrid.
"Right now that is the last thing that I'm thinking about", said Lopetegui when asked if he was anxious about losing his job.
"We have to sit down and think hard about what we are doing wrong and change our path".
Levante entered the game with the league's third-worst defence after conceding 14 goals in eight matches. The ball is just not going in. But the decorated France global, fresh from winning the 2018 FIFA World Cup, has struggled this season and was particularly culpable in this game.
Real conceded two early goals against Levante, with Jose Luis Morales scoring before Roger Marti converted from the penalty spot. But the more worrying aspect for Lopetegui is the results, something that his predecessor Zinedine Zidane used to grind out when in charge of the Spanish club.
Next up for Real is a Champions League game at home to Viktoria Plzen on Tuesday in which they will hope to gain some confidence before travelling to the Nou Camp on Sunday for the latest instalment of El Clasico against Barcelona.