Rajoy suggested that his first approach would be to try and form a minority administration.
Center-right People's Party made gains while anti-austerity party Podemos was left disappointed.
A series of polls on Sunday and Monday put 'Unidos Podemos' in second place overall.
'There's no alternative to Catalan independence. You'd have to be very radical to defend what we have now,' Puigdemont says.
The country's next government will need to show it is serious about fiscal policy.
The country's missed deficit target may mean spending cuts or tax hikes.
The anti-austerity firebrand will forgo a deputy PM role if it blocks a progressive government.
The country's ongoing political deadlock is taking its toll.
The country just wants a new government as soon as possible
After a strong result in a new poll, the party's hand is strengthened ahead of coalition negotiations.
'Winter is coming' and there is no new Spanish government in sight.
A look at all the options available to PM Rajoy, the King and politicians for Spain's new government.
Spaniards will cast their ballots on December 20 after a turbulent few years for Spanish politics.
The Podemos-backed local leaders have spoken of their support for a citizen solidarity register
Both parties voiced support for the movement, whilst organisers estimate at least 100 supporters attended.
Pablo Iglesias is on a mission to save Spain.
Pablo Iglesias made the challenge at his 'alternative state of the nation' address on Wednesday.
Crowds chanted "yes we can" and "tic tac tic tac", suggesting the clock was ticking for the political elite.
It will be the party's first major event in the Spanish capital.