The Toffees are facing tough competition for both but having proved a success last season – the Merseyside club have the upper hand.
Lukaku is at the World Cup with Belgium and has a number of suitors in England and around Europe – so if he impresses this month, Everton may be unable to attract the 21-year-old.
Barry is a free agent and has numerous offers on the table with the likes of Hull and West Ham keen, however Everton’s Europa League football should stand them in good stead.
“Lukaku we would love to keep. Gareth Barry we would love to keep and that is ongoing work and we obviously have to add to the squad, so it is non-stop,” chairman Bill Kenwright said.
“Romelu was a great addition to Everton. I have to tell you, off the pitch as well as on, he is an extraordinary boy. As you know he speaks all those languages, he’s got a terrific family and he walks the walk as well as talking the talk.
“We would love to have him back and people at Chelsea are good friends of mine so we are hopeful. But he is at the World Cup now and I think we will have to see what happens at the World Cup.
“Gareth Barry had a great season for us and I know the manager has spoken to him and I will meet with his people and we are hopeful that he will be there next season.”