Rajoy met with IOC President Jacques Rogge on Thursday to show his government's support for the Madrid bid, which picked up vital momentum a day earlier with its campaign pitch to the IOC general assembly.
Tokyo and Istanbul also made strong presentations, but it was Madrid that made the most impact with a show-stealing speech by Crown Prince Felipe, a former Olympic sailor and Spain's flag-bearer at the opening ceremony of the 1992 Barcelona Games.
Once viewed as a long shot because of Spain's severe economic crisis, Madrid is considered firmly in contention going into the final phase of the race before the Sept. 7 vote in Buenos Aires. The Spanish capital is bidding for a third consecutive time.
"We have very high hopes indeed," Rajoy said. "This time around, Spain is in a really good position."
Madrid finished third in the voting for the 2012 Olympics and second for 2016. Rajoy attended those votes in Singapore and Copenhagen, Denmark, when he was leader of the opposition.
Rajoy will travel to Argentina straight from the G20 summit in St. Petersburg, Russia, from Sept. 5-6.
"I am making the effort to go to Buenos Aires because that is the last push," he said.
Turkish Prime Minister Recep Tayyip Erdogan or President Abdullah Gul is expected to be in Buenos Aires for the Istanbul bid, with Japanese Prime Minister Shinzo Abe representing Tokyo.
The state of Spain's economy had been seen as a major weakness for Madrid's bid, with the country facing recession for most of the past four years and an unemployment rate of 27 percent. But Madrid says it already has 80 percent of its Olympic venues in place and would spend only $1.9 billion on construction for the games.
"The investment necessary to bring off Madrid's bid is much less than the other bids," Rajoy said. "Spain has already presented two bids. As a result of this process, almost everything needed for the 2020 Olympics is already built, is already up and ready to go."
Rajoy cited studies by the International Monetary Fund and other groups showing that Spain's economy is on the road to recovery.
"The Spanish economy is going through difficult times as is the rest of the European Union, but right now it's 2013," the prime minister said. "Next year the Spanish economy and the Spanish employment situation will definitely improve. ... In 2015, we're looking to be back on track for normal growth.
"There's only two months to go. I think the doubts about the economy, they really don't exist anymore because they're being cleared up by the data we're receiving."
Rajoy said he told Rogge the bid has the backing of all political parties and trade unions in Spain. He cited the IOC's own survey showing 81 percent public support for the bid.
"There's nothing, there's no one in all of Spain who has as much support as the Olympic Games do," Rajoy said. "I'd be happy to have half that number as prime minister."
As the three bid cities opened their exhibition rooms on Thursday to IOC members, showing off videos, maps, charts and other details of their Olympic projects, the talk among members was that Madrid gained the most from Wednesday's presentations.
"The overall perception is that Madrid was the best," Australian IOC member John Coates said. "It was a really powerful performance by the crown prince. He clearly believes in the Olympic ideals. It's not just talk."
While Tokyo was once considered the favorite and Madrid an afterthought, the dynamics may have shifted after Wednesday's performance by the Spaniards.
"It might have got them in the race," Coates said. "I think it's remarkably tight."