The 33-year-old has spent the 2013/14 campaign on loan at Goodison Park from Manchester City.
His contract with his parent club is due to expire this summer, dropping him into the free agent pool.
Everton are eager to snap his experience and talent up on a permanent basis, but are likely to face competition for his signature.
A move abroad has been mooted and Roberto Martinez concedes that only Barry can decide what his footballing future holds.
The Everton boss said in the Daily Express: "We will talk after the Hull game. I'm sure we will get a straight answer in one way or another.
"It's not about finding a good contract or financial package, it's more about the next footballing chapter and where he can enjoy it.
"He has loved his time at Everton but you don't know where his next step is. He may want to go abroad and experiment with a different culture.
"It's important we try to keep him at the club but in the same way if he doesn't stay, it's our job to make sure the next squad in the next window is as strong or stronger than what we have.
"I learned many seasons ago that you can't waste any energy on the ifs and buts. It is more about what you can affect. You need to put things in place and make sure the options ahead are exciting and that is where we are as a club."
Whatever the long-term future holds for Barry, Martinez believes that the short-term should include further international recognition with England at the World Cup finals in Brazil.
The Spaniard added: "I would take him. He has been so consistent throughout the season and his level of performance has been phenomenal.
"What surprised me more than anything was how he developed a new partnership with James McCarthy. I've been so impressed with that."