Los Merengues have been searching for a defensive midfielder since the end of last season, as they tried to Lure Liverpool's Xavi Alonso and Daniele De Rossi of Roma.
However, talks with the two clubs were not fruitful and Real are now in the hunt for D'Agostino, who can be a much easier and cheaper target than Alonso and De Rossi.
"There are negotiations in place and we will see if there are further details and other meetings soon," Udinese chairman Giampaolo Pozzo told Sportmediaset.it.
D'Agostino was on the verge of signing for Juventus last week but the deal collapsed when the two teams couldn’t agree on a price for the 27-year-old.
"The player's price will not change. He costs €25 million for Real Madrid like it was for Juventus," Pozzo added.
D'Agostino previously played for Messina, Bari and Roma before excelling for Udinese this season to earn himself his first ever Azzuri call-up in November 2008.
He made one appearance for Marcello Lippi's men so far when he played against Northern Ireland on June 6 this year.
Watch D'Agostino's skills