Relegation-threatened Livorno have sacked Domenico Di Carlo after just 88 days in charge and re-appointed Davide Nicola.
Virgil van Dijk claims he knows nothing of interest from English clubs and is keen to prove himself in the Champions League with Celtic.
The Spanish stopper put pen to paper on a deal that will keep him in Sardinia until the end of the season with an option to extend the contract for a further year, the Italian club confirmed on Wednesday.
The 26-year-old, who came through the Real youth ranks, became a free agent in June after the Madrid outfit failed to offer him a contract extension.
"I am delighted to be here," Adan said to cagliaricalcio.it.
"I wish to thank club president Massimo Cellino and the team for giving this opportunity."
Cagliari are 13th in Serie A and having given away 20 goals in 12 games.
The Sardinians travel to the Italian capital on Monday to face league leaders Roma.