The 27-year-old Colombia forward was undergoing a medical in Turin ahead of an anticipated season-long loan and whether there is an option for the move to become permanent remains to be seen.
He moved to Stamford Bridge from Fiorentina in February, but made a limited impact as the Blues won the Capital One Cup and Barclays Premier League title.
"We all know that Juve is a great club, among the best teams in Europe
They have everything to win the Scudetto (Serie A)," Cuadrado said on the club's official website.
"I hope I can give Juventus my best, 100 per cent
I didn't expect to receive such a warm welcome from the fans in Italy but they have always welcomed me with open arms and that makes me very happy.
"The fans' support will give me further motivation to do well."
Cuadrado was one of Colombia's star players at last year's World Cup but has failed to make an impact at Stamford Bridge since moving from Fiorentina for an initial £23.3million.
He signed a four-and-a-half year contract at Chelsea but it seems the club are already willing to allow him to play elsewhere after four Premier League starts.
It will inevitably lead to suggestions France midfielder Paul Pogba will make the switch in the opposite direction, with a move in excess of £60m spoken of.
Chelsea boss Jose Mourinho refused to discuss Pogba last Friday and the Cuadrado switch appears independent of any negotiations for the former Manchester United midfielder.
Whether anything materialises over Pogba remains to be seen, but the rumours remain.
Cuadrado is excited at the prospect of playing for Juve, who won the Italian double - Serie A and Coppa Italia - in the 2014-15 campaign and finished runners-up to Barcelona in the Champions League.
He is eager to win another league title.
"The league has only started and we hope we can defend the title," he added.
"I want to prepare well for every game and then the coach will decide
I just have to work hard and play my game."
Juve, coached by Massimiliano Allegri, began their Serie A title defence with a 1-0 defeat at home to Udinese on Sunday and take on last season's runners-up Roma at the Stadio Olimpico this weekend.
Chelsea's Egypt winger Mohamed Salah, who moved to Fiorentina on loan as part of the Cuadrado deal, is now on loan at Roma.
Source : PA
Source: PA