The eurozone's fourth-largest economy has had only a caretaker government since the December 20th general election produced a hung parliament in which a ruling majority cannot easily be formed.
Rajoy's conservative Popular Party (PP) won the most seats in parliament, 123 of 350 seats, but fell short of an absolute majority.
"It is obvious that uncertainty has consequences. This is why it is so important that we concentrate on what matters... and the country needs a government as soon as possible," he said in an interview with Spanish public radio.
Rajoy has called for a "grand coalition" of the PP, the Socialists who came in second place winning 90 seats and new centre-right party Ciudadanos which took 40 seats in Spain's lower house of parliament.
But he has so far won implicit support only from Ciudadanos.
The Socialists have rejected the PP's advances and appear to favour a left-wing alliance that includes new anti-austerity party Podemos which won 69 seats along with small, regional nationalist parties.
"No is no," Socialist leader Pedro Sanchez said Monday as he reiterated his opposition to helping Rajoy form a government.
But Sanchez's desire to form a left-wing coalition with Podemos is complicated by the new party's support for an independence referendum in the rich northwestern region of Catalonia.
King Felipe VI on Monday began a week of meetings with political party leaders before nominating one of them to try to form a new government.
The nominated party leader must win a parliament vote of confidence to take office.
The PP wants to hold a first vote on its proposal for a government during the last week of January or the first week of February.
The king traditionally has invited the winner of the most seats to form a government but he can opt for other leaders if it appears that they are better positioned to provide a stable cabinet.
That has happened at the regional level in Spain but never following a general election.European Commission chief Jean-Claude Juncker said Friday he hoped Spain would form a "stable government as quickly as possible".
"From the point of view of companies, what matters to us is that there is a solid, strong government formed by those who will face up to economic difficulties," the head of the Spanish Confederation of Business Organisations (CEOE), Juan Rosell, told AFP.