The Dutchman avoided the sack after he led Man United to FA Cup glory in May's Wembley final against Manchester City.
Ten Hag was subsequently rewarded with an additional one-year on his deal to extend his contract until the summer of 2026.
However, he is already facing another battle to keep his job after seeing his side win just two of their opening seven Premier League games.
With Man United down in 10th position, there is speculation suggesting that the Man United hierarchy could turn to former Bayern Munich boss Thomas Tuchel.
Rooney backs "fantastic manager" Ten Hag
The Man United hierarchy are expected to discuss Ten Hag's future in a meeting on Tuesday, but the Dutchman is still confident that he will remain in the Old Trafford dugout.
Ahead of that meeting, Ten Hag has received the support of ex-Man United star and current Plymouth Argyle boss, Rooney.
When asked about whether Ten Hag was the right man for Man United, Rooney told Sky Sports News: "Well that's obviously down to the owners of the football club to look at that. I'm sure Ten Hag is a fantastic manager.
"He's won trophies wherever he's been and won trophies at Man United as well. So yes, he's going through a difficult period and I think he'd probably say that himself.
"But you hope he can turn it around because too often managers lose their jobs and then it's another rebuild with a different manager."
Rooney sends message to Man United squad
While Ten Hag is under scrutiny, Rooney believes that the players also need to take some responsibility for the club's current predicament.
Rooney added: "I think it's difficult and anyone would admit that Manchester United want to be challenging for titles, so to be 14th, I'm sure everyone at the club knows they need to improve on that.
"I know Ten Hag is doing everything he can to try and improve that. Players again, and I said this last season, need to do better than what they're doing. There is a big turnaround of players over the last few years. You just want to see them challenging again.'
Man United will resume their Premier League campaign on October 19 when they welcome Thomas Frank's Brentford to Old Trafford.