Prime Minister Pedro Sanchez's Socialists won elections in April but fell far short of a majority.
Sanchez now has until September 23 to be confirmed as premier by a deeply fragmented parliament — or face fresh elections on November 10.
Polls suggest the Socialists would win more seats in such polls but still fall short of a majority.
Sanchez has failed to reach agreement with the far-left Podemos, whose support is key along with that of several smaller regional parties.
Seeking to break the logjam, the monarch on Thursday met with Speaker Meritxell Batet ahead of talks next Monday and Tuesday with party parliamentary leaders.
The king will try to “propose a candidate for the head of government who can count on the necessary support to win a (parliamentary) vote of confidence,” a palace statement said.
Failing that, the parliament would be dissolved and new elections would be called, the statement said.
Sanchez was expected to receive a fresh mandate from the king to once again present his candidacy as head of government to parliament.
The Socialist leader came to power in June 2018 by winning a surprise no-confidence vote against conservative predecessor Mariano Rajoy with the support of Podemos as well as Catalan separatist parties and Basque nationalists.
But he failed twice in July to be confirmed by the assembly following a dispute with Podemos and its leader Pablo Iglesias that has yet to be resolved, with the two men trading accusations over the deadlock in parliament on Wednesday.
Podemos wants to be part of the cabinet, but the Socialists have refused, offering only talks on a joint policy programme.
In July, following lengthy negotiations, the Socialists had offered Podemos a number of government posts but Iglesias turned them down, saying they did not carry any political clout.