Alfredo Perez Rubalcaba said Mr Rajoy “cannot lead the country” as Spain faces a huge economic crisis.
El Pais has published images of ledgers showing payments to members of the governing PP. Mr Rajoy says they are false and denies receiving payments.
Protests against Mr Rajoy were held on Saturday in several cities.
An online petition demanding the leader’s resignation has gathered more than 740,000 signatures in just 3 days.
Mr Rubalcaba said Mr Rajoy should “cede his place to another leader”.
“His presence is not going to help resolve this political crisis,” he said.