To succeed, the no confidence vote will need the support of an absolute majority of 176 lawmakers, a difficult task as opposition parties are deeply divided.
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While far-left party Podemos has already called for a no-confidence vote, the motion will need the unlikely support of centrist party Ciudadanos, Rajoy's allies in parliament, or that of several tiny regional parties, including Catalan separatists.
The PSOE holds 85 seats in Congress, so they will have to garner a further 91 votes for the no-confidence vote to be successful.
Podemos leader Pablo Iglesias, whose central party and regional branches control 71 seats in Congress, confirmed that he would support Sánchez if he tabled a motion of no-confidence.
"No advanced democracy can put up with a party of criminals in government", he said.
Which means the PSOE needs another 20 seats to effectively oust Rajoy.
Ciudadanos, led by Albert Rivera, has 32 seats but has supported the PP in government, helping to pass a difficult budget, but has been outspoken on corruption and the need to clean up politics.
But Ciudadanos, which has seen a meteoric rise in the polls, has little common ground with the PSOE and even less with Podemos, and all would have to agree a replacement candidate for PM, which is likely to be Pedro Sanchez.
If Ciudadanos refuse to support the no confidence vote then the PSOE must rely on the regional nationalists, the Basque PNV , which has five seats, and the more troublesome Catalan nationalist parties—Republican Catalan Left (Esquerra, ERC) and Convergencia—who together hold 17 seats.
In fact, Ciudadanos was quick to dismiss support for a no confidence motion and instead demanded that new elections be called.
Jose Manuel Villegas, the secretary general of Ciudadanos in a press conference on Friday said that he believed Sanchez would “rely on the support of those who want to break up Spain” in his quest to reach Moncloa and that Ciudadanos would not support that.
📽️ @CiudadanoVille "Si el señor Rajoy se niega a llamar a votar, apoyaremos una moción de censura; o convoca elecciones o lo hará el Congreso mediante una moción instrumental" #ActualidadCs pic.twitter.com/1mKycqCJGC— Ciudadanos (@CiudadanosCs) May 25, 2018
Instead he said that the motion of no confidence should be withdrawn and the PSOE should join the push for new elections.
New elections will favour Ciudadanos, after the PP has seen its popularity plummet even before the latest graft ruling.
According to a Metroscopia survey conducted for El Pais in early May, Ciudadanos would be the clear winner in new national elections, with 29.1% of the vote.
Second place would go to Unidos Podemos, an alliance between Podemos and other leftist parties, with 19.8 percent.
Support for the PP would slip to 19.5 percent, while the PSOE would come in last with 19 percent of the vote.