The 24-year-old had been linked with a number of sides during the close-season but opted to swap the Eredivisie for La Liga at the last minute and signed a four-year deal with Atletico.
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"The club have been following me now for two or three weeks, and you could see they really wanted me," he told reporters.
"That is why I turned down approaches from other teams. Liverpool, for example, appeared in the last few days, but I had already made my decision.
"The truth is that Atletico made a huge effort to convince me."
Ilori and Sakho, who arrived from Sporting Lisbon and Paris Saint-Germain respectively, helped to bring Brendan Rodgers' tally of new additions up to nine - with Chelsea star Victor Moses also completing a late loan move.