Sanchez has repeatedly refused to enable a minority government led by Rajoy's centre-right People's Party (PP), which won the most votes but fell short of a majority in national elections in December and in June.
Expectations have risen in recent weeks that the Socialists would eventually abstain in confidence votes to allow Rajoy a second term in office and avoid third elections, but Tuesday's meeting yielded no breakthrough.
"The right-wing parties are the ones that have to reach an agreement, the left is not going to support the right," Sanchez told a news conference.
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(Reporting by Sarah White and Angus Berwick, Editing by Paul Day)
