Izaguirre claimed on Saturday that Manchester United had made an approach for him, and Kayal has been linked with Liverpool and Valencia.
After signing a new 12-month contract himself last week, Lennon is now looking to tie down some of his key players and revealed that talks with Kayal had begun ahead of their departure for Australia this week.
"We've opened preliminary talks with Kayal's people," the Northern Irishman told the Sunday Scottish Mail.
"He knows I want to keep him and he's making the right noises about staying. That's great news. It's the same for Izaguirre.
"Going to Australia on tour this week will give me the chance to sit down with Beram.
"I'll need to sit down with Beram, look him in the eye and get a feel for what he's thinking. He knows I have his best interests at heart."
Lennon believes both players will benefit from remaining at Celtic Park, although he admits keeping them would be difficult should a top-four Premier League club come knocking.
"I can help them progress. They have the supporters on their side and a great affiliation with them. They would struggle to get that at many other clubs," he added.
"I'm sure there are Premiership clubs that could offer them better deals but at this stage of their careers money shouldn't be that important.
"There is no need for them to rush off and go to the Premiership just yet. They can stay here a year or two and then look at things after that.
"My whole point to them will be that unless it's a top-four club in England then I don't think there are many better places to be than here."