Real are looking to strengthen the centre of midfield following their spending spree on the likes of Portuguese winger Cristiano Ronaldo and Brazilian playmaker Kaka and may also sign another full-back, Pellegrini said at a news conference.
"It's not a mystery that Real Madrid are interested in signing him (Alonso) but it won't be easy," he said.
"We are looking at alternatives for some positions," he added. "The attack is complete and perhaps a central midfielder or full-back is still missing but the rest of the positions are covered."
In his first news conference since the players began pre-season training last week, Pellegrini also said media reports that Real's entire contingent of six Dutch players, including Ruud van Nistelrooy, Arjen Robben and Wesley Sneijder, would be leaving were inaccurate.
"When there is some news the club will announce it," he said when asked which players were on their way out.
"But it's not true that all the Dutch players are going to leave as has been reported in the media."
Pellegrini also responded to criticism of Real's spending policies published in Vatican newspaper Osservatore Romano on Saturday.
The club's splurge of more than 200 million euros on four players risks inflating transfer fees and pushing football clubs towards bankruptcy or into the hands of organised crime, the paper said.
"Everyone has the right to give their opinion, right or wrong," Pellegrini said.
"I am not the one responsible for talking about comments by the Vatican. If someone thinks that it's not good they also have the right to express an opinion."
Separately, Kaka said in a newspaper interview published on Sunday that Real were spending such vast sums because the players involved were value for money.
"If Real Madrid has decided to spend so much money on us it's because we are worth it, no?" Kaka told sports daily Marca.
"In the particular case of Cristiano Ronaldo, who is the most expensive, he has shown he is number one and that he is worth the money."
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