Both enjoyed fruitful loan spells with the club last season and manager Roberto Martinez has declared his intention to sign them both this summer.
Barry is a free agent after his Manchester City contract expires and the fact he hasn't signed up yet suggests he is considering his options, perhaps more lucrative ones.
Lukaku meanwhile seems to be linked with a different club each day, with the likes of Atletico Madrid, Wolfsburg and, er, West Ham linked with the striker this month.
Both signings would be a real coup for Everton, especially Lukaku who at only 20-years-old can only get better (and more valuable).
Kenwright certainly speaks well and makes his intentions clear, but as a former actor that is hardly surprising. What he now needs to do is back up those words with action, something he hasn't always done while in charge of Everton.
Still, the very fact these quotes have emerged, especially the revelation that he has a close relationship with some of the senior management at Chelsea, gives us reason to be positive.
However, we have been let down too many times in the recent past before to get excited. This summer's transfer dealings will almost certainly go all the way to the deadline - again.
"Lukaku, we would love to keep. Gareth Barry, we would love to keep. And that's ongoing work," said Kenwright. "We obviously have to add to the squad, so it's non-stop.
"Gareth Barry had a great season for us. I know the manager has spoken to him and I will meet with his people and we are hopeful he will be here next season.
"Romelu was a great addition to Everton, I have to tell you, off the pitch as well as on. He is an extraordinary boy. He speaks all those languages, he has 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's at the World Cup now and we will have to wait and see what happens at the World Cup."