The 26-year-old stopper has made 47 league appearances for the Israeli outfit since joining in 2007, scoring four times from the penalty spot.
And the Nigerian international - who has 43 caps for his country - was believed to have been close to completing a move to Arsenal, only for Hapoel's demands of €2 million for his services causing negotiations to break down.
However, fellow English outfits Hull and Wigan are now thought to be weighing up bids for the glove-man, and Enyeama remains hopeful that a deal can be concluded in time for the start of the new season.
"I am calm. There are contacts in the Premier League," he told skysports.com.
"I am interested in playing there next season but nothing is done yet.
"I want to prove I am the best in the world, so I need to face world class strikers week-in, week-out."
NEWS: Arsenal came close to signing Enyeama
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