The Netherlands goalkeeper has not played for Fulham since the 2013-14 campaign, in which they were relegated to the Championship.
Koeman rates his compatriot highly, though many of his 17 Premier League starts for Southampton last season were made while first-choice Fraser Forster was injured.
Nevertheless, Get West London reports that the veteran could well find himself at Everton working under Koeman again, with the Merseysiders in need of a goalkeeper following Tim Howard's departure to the United States.
They signed Chris Renshaw from Oldham Athletic this week (Sky Sports) but, at 18, the teenager is unlikely to feature prominently next term.
By virtue of the fact he turns 34 in September and his contract expires next summer, Everton would not have to pay a lot for the shot-stopper, if indeed the report is accurate.