Williams was brought in by Koeman last summer to shore up the Everton defence and has formed a near-permanent partnership with Ramiro Funes Mori at the hear of the Everton back four.
Now, he wants Koeman to continue the job he started at Goodison Park when he left Southampton for Everton last summer.
“For me personally and the lads, we obviously hope he stays,” said Welsh defender Williams.
“He’s started building a team here and we’re all confident the future looks good for Everton.
“But he might have his own ambitions. I can’t speak for what he wants from his managerial career but from our point of view, definitely we enjoy working with him.”