The City manager is now inside the final year of his current deal.
And Begiristain said in the new book, 'The Pep Revolution': "It’s something I think about from time to time but it’s not easy to come up with the right person.
"There are coaches whose style of football or methodology I admire. Guys who get great results. But it’s very difficult to find someone who even gets close to Pep.
"We’ve still got another year so we’ve got a bit of time to keep thinking about it. The best thing would be to hold on to Pep. I’m well aware that that’s a tough ask. By 2021 he’ll have been here for five years and that’s already much longer than he’s spent at any other club, but it would definitely be the perfect solution.
"If we can persuade him to renew with us again. Let’s see if we can pull it off.”