The reports said Calderon, who has denied any wrongdoing, was pushed into resigning by several members of the Real management board and would be replaced by Vicente Boluda, one of the club's vice presidents.
Asked about the reports, which did not identify any sources, a Real Madrid spokesman said the club had no immediate comment.
The AS newspaper said the board would meet at midday on Friday to analyse the situation and some members wanted fresh elections to be held, possibly on May 24.
Calderon, 57, has been under pressure since Marca reported this week that unauthorised persons gained access to the club's assembly on December 7 and their presence had helped swing votes on key issues in favour of the management.
At a news conference called to address the allegations on Wednesday, Calderon said two Real officials involved in organising the assembly had resigned and he would probably not stand for re-election when his four-year mandate ended in 2010.
He ruled out stepping down immediately and insisted neither he nor any other member of the management board were involved in any irregularities.
"None of us participated either directly or indirectly in these acts," he said. "We did not do anything. We are victims."
FAILED BIDS
Real have won back-to-back Primera Liga titles but Calderon has still been criticised by fans and local media since taking over in 2006, often for failed bids to bring in high-profile players.
His woes have sparked speculation that former president Florentino Perez may be planning a comeback.
Perez resigned in 2006 after his six-year experiment in assembling a "galactico" super team collapsed as the club headed for a third successive season without a major trophy, their worst run in more than 50 years.
Calderon has been unable to make good on a pledge to bring Arsenal midfielder Cesc Fabregas and AC Milan's Brazilian playmaker Kaka to the club and his attempt to lure Manchester United winger Cristiano Ronaldo also ended in failure.
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