The leader of Unidas Podemos, Pablo Iglesias, launched a new offer of a coalition government to the leader of the PSOE, Pedro Sánchez, on Wednesday.
Sánchez needs the support of the MPs of left-wing Unidas Podemos to be voted in as prime minister in a forthcoming debate, now set to start on 22 July.
Iglesias said he would be happy to go into coalition and now wouldn't need any Cabinet-level ministerial positions for his party.
The Socialists replied that they would only accept a "government of cooperation" and not a coalition government with Unidas Podemos.
Sánchez will meet with leaders of the different parties again next week to try to find a way to get the votes or abstention he needs.
In the event that Sánchez fails to be appointed in investiture votes in July, or at the latest September, an opinion poll has shown this week that he would win more seats if the general election has to be held again in the autumn.