Enyeama, who has one year left to run on his contract with Israeli champions Hapoel Tel Aviv, says he expects things to move quickly within the next week.
"I expect to hear something in a week's time," he told KickOffNigeria.com as the Super Eagles headed for home from South Africa after a dismal performance in which he came out as the brightest star.
"It is a normal thing. We all want to make progress in life. So it is natural for me to want to get a move to somewhere else."
A slew of clubs have reportedly been on the queue to sign him up, but the Nigerian says he is not aware of any interest of yet.
"I turned my phone off during the World Cup because I did not want to be distracted from representing my country. So I have not heard anything. Now, I just want to go home and spend time with my family so we can all thank God for what he has done for us. Then I have to go back to Hapoel to prepare for the Champions League qualifiers and we will see what happens."
Enyeama was offered an extension by Hapoel before the World Cup, but turned it down, and says he has no particular preference on which country he would like to move to.
"I just want a good league where I can face the very best players in the world every week."