The Colombia winger, 26, has already agreed personal terms with the Premier League leaders and is expected to have his medical in London on Friday.
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The fee could rise to £26.8m - Cuadrado's release clause.
Mohamed Salah, 22, is likely to go on loan to Fiorentina as part of the deal, with another winger - Andre Schurrle - in talks with Wolfsburg.
Montella said: "It was a sale that had to be done.
"The club has done the right thing because we are also getting a very good player. Naturally, we need to understand how long Salah will need to adapt to our football as it's different from the English and Swiss leagues.
"From a defensive standpoint, Cuadrado guarantees us a lot more than Salah. Cuadrado is a player that has always arrived with a smile and given everything to Fiorentina.
"I am convinced that if he could have played tomorrow against Genoa he would have. I'm happy for him because he is going to a big club."
Germany forward Schurrle, 24, is almost certain to leave before Monday's transfer deadline. Wolfsburg have opened discussions on a potential £22m deal.
"James Rodriguez was Colombia's undoubted World Cup star - but Juan Guillermo Cuadrado was not far behind.
"Chelsea's new acquisition solved Colombia's tactical problem. The defensive line had to sit deep to protect veteran centre back Mario Yepes.  It was Cuadrado who kept dragging the team forward and opening up the pitch with his sustained pace and effervescent dribbling down the right flank.
"He is at his best as a wide right-midfielder - especially as over the last couple of seasons he has started scoring more goals. But he can also feature as an attacking right-back, or cut in from the other flank.
"Some three years ago a Premier League giant had a look at Cuadrado but concluded that he was too old - a strange decision given that he is still only 26, and should be looking forward to playing his prime years under Jose Mourinho, who will certainly enjoy the pace on the transitions that his new Colombian acquisition will bring to the side."
Chelsea are set to complete the £23.3m signing of Juan Cuadrado, Fiorentina coach Vincenzo Montella has said.