"Spain deserves a debate between you and me face to face on TV, whenever and wherever you want," Podemos leader Pablo Iglesias told a party gathering at a Madrid theatre.
Iglesias, a 36-year-old pony-tailed former university professor, painted a dark picture of the Spanish economy a day after Rajoy defended his government's record during an annual state of the nation address in parliament.
"Come with me, descend to see the reality of my country," said Iglesias.
Under Rajoy salaries have dropped, the unemployment rate stands at over 23 percent, the highest level in the eurozone after Greece's while the vast majority of jobs that were created last year were short-term contacts, Iglesias said.
"According to the tax office, there are nearly 7.7 million workers who earn less or much less than a thousand euros a month," he added.
Rajoy said during his state of the nation address that the "nightmare was over" for Spain after the country returned to growth last year of 1.4 percent, the first full year of economic growth since a 2008 property crash which threw millions of people out of work.
"The nightmare is the reality of our country," Iglesias said during his speech in response to the prime minister.
Podemos was set up in January 2014 and his no representation in parliament but it regularly comes in either first or second place in opinion polls, ahead of the main opposition Socialists.
Andalucia, Spain's largest region, will vote in a regional election on March 22, with local and regional elections in the rest of the country slated for May 22 and a general election expected at the end of the year.