The trip followed the French club’s home defeat by Lorient on Sunday, in which Messi played the full 90 minutes.
Messi will not train or play for PSG during the period of his suspension.
It is understood the 35-year-old asked permission to make the journey to carry out commercial work but was refused.
Messi, who has also been fined by the club, has a role as a tourism ambassador for Saudi Arabia.
The World Cup winner’s two-year contract with PSG expires this summer.
Barcelona vice-president Rafael Yuste claimed in March that the Spanish club were in contact with Messi about a return to the Nou Camp.
Messi has scored 31 goals and contributed 34 assists in 71 games in all competitions for PSG, and won the Ligue 1 title last season.
He is set to miss matches against Troyes and Ajaccio as PSG, five points clear with five games to go, look to clinch a ninth league title in 11 seasons.
Messi has taken a decision that effectively calls time on his Paris St-Germain career.
Yes, they have three games left after Messi’s suspension has been completed and there is work remaining to secure another Ligue 1 title, but PSG are on a different course now – and it does not involve Messi, who less than five months ago achieved the crowning glory of his stellar career by lifting the World Cup.
PSG do not view their actions as being anything extraordinary. In their minds they are effectively punishing an employee who has gone somewhere else on a work day miles away from where he is supposed to be.
But they also feel it is a statement about the future direction of the club, which they are adamant will be around younger players. It is also confirmation of their zero-tolerance approach to discipline.
PSG’s fans don’t want Messi any more. It is certain his contract will not be renewed.