New Barca manager Ronald Koeman decided that the pair should train individually, accompanied by a fitness coach, rather than join the rest of the squad.
Both players were informed in brief conversations with Koeman last week that they were not part of his plans next season as Barca look to rebuild their squad after a disastrous 2019-20.
Suarez, who has risen to third in Barca's all-time goalscoring charts since joining in 2014, is expected to move to Turin for a nominal fee.
He is now likely to link up with his former coach at Juventus, Antonio Conte, in a return to Serie A.
ESPN reported that Koeman has identified Liverpool's Georginio Wijnaldum as the player he would like to replace Rakitic and Vidal, as well as summer signing Miralem Pjanic and youngsters such as Riqui Puig.
Koeman -- who replaced Quique Setien as Barca manager after their humiliating Champions League exit -- has now completed his first week of training sessions with the squad.
His idea is that players who will not feature for the team next season should step aside now, so he can focus on younger players who will be competing for opportunities in 2020-21.
Meanwhile, former Barca boss Luis Enrique said both the club and Messi will be successful when they part ways one day.
"Barcelona are one of the biggest in the world and have won many titles. It's clear they have had a marvellous relationship. Leo has made Barca grow exponentially thanks to his great performance. I would have liked there to be an agreement.
"Sooner or later, Messi will stop playing at Barca. It would have been good if it was more friendly. It will be a pity, the day he goes, we will not see that marvellous player, but the club will keep winning titles without Messi, just as Messi will keep being marvellous for many years if he goes."