Hodgson included seven players under the age of 23 in his final squad for the tournament in France, where England can reach the last 16 with a point against Slovakia on Monday.
"It is down to his bosses but of course, as players, we want him to stay," Rooney, who will reportedly be left out of the starting lineup on Monday, told reporters.
"It shouldn't go unnoticed the amount of young players he's introduced into the squad over the last four years. There was a big issue in terms of everyone saying we didn't have the young players coming through in England which other countries had."
Wales, a point behind England in Group B, play Russia in Monday's other game.
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(Reporting by Ian Rodricks in Bengaluru; Editing by Peter Rutherford)
