The former Middlesbrough manager has made a decent fist of steadying the ship after Sam Allardyce's sensational exit, overseeing three matches without defeat while attempting to implement philosophies more fitting with the times, such as building from the backline.
"Southgate has done brilliant with the Under-21s and he's come in and done a really good job," Rooney told Sky Sports' Mark Ashenden.
"He's got two wins from three and a good point from the last game in Slovenia. It's down to the FA now and he's put himself in pole position and I'm sure the right decision will be made."
Southgate told reporters that he needed to "reflect" before considering his England candidacy following the win over Scotland, but was pleased with what he saw in the Wembley wrangle.
"We caused our own problems (at the back) but once we were through that phase on the field some of our use of the ball was fantastic," said Southgate post-match.
"The team in an attacking sense played well. We played with an interchange of positions and there were some superb individual performances and we managed the game well towards the end."
England has a friendly visit from Spain in Tuesday's spate of international fixtures.