On Tuesday, Mainz sporting executive Rouven Schroder had said that Szalai, a star at the club in the early 2010s who returned in 2019, should look for a new club.
Szalai, 32, subsequently showed up for training the following day but was told to train away from the senior squad, which led to his teammates not taking the pitch for the afternoon session.
Speaking at a news conference on Thursday, originally scheduled as a prematch news conference for the Bundesliga match against Stuttgart, the club's leadership denied the incident was related to Szalai's role as a team representative demanding outstanding salaries, held back during the first phase of the coronavirus pandemic.
"What happened yesterday, for us all, who love football, was an awkward situation," Schroder said.
"We know that the Adam Szalai decision would not be popular. We made a wrong judgement about the way of communication and it was charged because of aggressive media coverage too."
Schroder added that the squad will need to return to training and insisted Szalai had been cut from the squad for "sporting reasons" only.
In a statement released on Wednesday, Mainz had said: "The pro team's training session on Wednesday afternoon could not go ahead as planned.
"The cause for it were emotional discussions of the squad regarding Adam Szalai, who was told not to train with the senior squad in the afternoon."
Amid reports of either a rift between the squad and coach Achim Beierlorzer or a clash over the payment of outstanding salaries not paid yet because of the coronavirus crisis, the club plan to address the issues at a news conference on Thursday.
Mainz began their Bundesliga campaign with a 3-1 defeat at Leipzig on Sunday and host Stuttgart on Saturday.