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 club say they have paid a "substantial" undisclosed fee for the 27-year-old, who has been on loan with the Millers since October and scored four goals in 16 appearances.
Frecklington has put pen to paper on a two-and-a-half year deal after the two clubs agreed a fee earlier this week.
The transfer is subject to Football League and FA ratification.