The 68-year-old has been in charge for four years and his contract expires at the end of this tournament.
FA chairman Greg Dyke told the BBC's Sportsweek that Hodgson's deal would be discussed after the Euros but would be renewed if England "do well".
"It is down to his bosses but of course, as players, we want him to stay," said Rooney.
England play Slovakia in Saint-Etienne on Monday knowing a point will ensure their place in the last 16 in France, with Hodgson reportedly planning to make six changes to his starting line-up from the team that beat Wales 2-1.
Captain Rooney is expected to be rested, along with Harry Kane, Dele Alli, Raheem Sterling, Kyle Walker and Danny Rose. Daniel Sturridge, Jamie Vardy, Jordan Henderson, Jack Wilshere, Nathaniel Clyne and Ryan Bertrand are likely to start.
