Koeman was tasked with telling Suarez that the club wanted to get rid of him, but the Uruguay international feels the Dutch coach hid behind the board's decision to force him out.
"Koeman's words were: 'They [the directors] told me to tell you,'" Suarez told radio station Cope on Monday. "I was made to train separately [from the first team] and left out of the squad for the three friendlies.
"Then, Koeman said that if my future was not resolved, he was counting on me for the first game of the season against Villarreal. That is when you see that he didn't have the character to tell me that he didn't want me himself."
However, Suarez acknowledges Koeman is not to blame for the way his departure played out after six successful years at Camp Nou.
At the time, Lionel Messi launched a scathing attack against Barca's board, accusing it of disrespecting his teammate. Suarez also laid blame at the feet of the former president, Josep Maria Bartomeu.
"Firstly, the president," Suarez added when asked who had treated him badly at Barca. "I will always be grateful to the club, but they used me. He said everything in the press instead of phoning me personally to say the club didn't want me.
"Why did he not call me directly? They phoned when they wanted me to convince [Messi] to stay and to speak with [Antoine] Griezmann [about signing]. They should have explained the situation to me."
Suarez said he had not received any messages from Koeman or Bartomeu congratulating him for firing Atletico to their first La Liga title since 2014, although he joked he was close to sending them a photo after winning his fifth league in seven years in Spain.
Prior to scoring the goal against Real Valladolid that clinched the title for Atletico on Saturday, Suarez had lunch with Messi in Madrid.
Messi is out of contract in June and has not yet revealed if he will sign a new deal with Barcelona. Suarez said his advice would be to stay in Catalonia, laughing when asked if Messi had already told him his plans.
"As a friend and a fan, I would be delighted if he stays at Barcelona," he smiled. "And that's what I would recommend."
Suarez, who for years had actively encouraged Barca to bring in a No. 9 to provide competition for him, feels it's a good signing for the club.
"It's a different board now, new people in charge and they're looking for players with quality," he said. "You can't argue with Aguero in that sense. I am in complete agreement with the signing."
Earlier this week, Suarez confirmed he would remain at Atletico for the second year of his contract following speculation he could leave this summer.