Chelsea striker Lukaku and Manchester City midfielder Barry both played key roles during loan spells last season as Everton finished in fifth place in the Premier League table.
Lukaku, 21, who is currently at the World Cup with Belgium, continues to attract the interest of a host of clubs despite having two years remaining on his Chelsea deal, while Barry, 33, is now out of contract after his deal with City expired.
Kenwright said: “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.
“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.”
Young Spanish forward Gerard Deulofeu has returned to parent club Barcelona following his loan stay with Everton last term and, if Roberto Martinez fails to bring him back to the club, then Kenwright is confident his manager will find an equally talented replacement.
“Who’s to say what will happen to Deulofeu next season but there will be other Deulofeus that he will be looking at,” he said.