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Defender Ramos is searching for a new club after leaving Real upon the expiry of his contract following 16 seasons at the Bernabeu.
The 35-year-old has been strongly linked with PSG, but Eto'o believes the Camp Nou would be an ideal - if somewhat controversial - destination.
As a PSG fan, Cameroonian Eto'o would also like Ramos to move to the French capital, but he suggested that Barcelona ought to make a move for the Madrid legend as they look to earn an elusive first Champions League title.
Eto'o, who played for both Barca and Real, told AS: "Hopefully Sergio goes to PSG. If I could sign him for Barca, I would.
"If he goes to PSG, I will be happy and I know that he will help us win the Champions League, which is proving very difficult for us."
Ramos picked up five red cards with Real's rivals over the years, but he also scored some memorable goals.
However, he struggled for fitness last season after making just 15 league appearances.
That lack of game time saw Luis Enrique leave Ramos out of his Spain squad for Euro 2020.
"It was a very difficult decision but he [Ramos] has not been able to play this season and has hardly trained since January," Enrique said.
"I called him - it was hard, difficult.
"I feel bad, because he is very professional, and he helps the national team a lot. And he can still help it in the future. But I have to look for the best for the squad."
Ramos broke down in tears in an emotional Madrid farewell earlier this month.
"This is not a goodbye, it's a see you later," he said.
"Because I will be back."
It is thought that PSG are Ramos' favoured destination.
Featured Image Credit: PA
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