"First of all you have to give Ole an opportunity and that's a discussion that has to take place with the Glazers," said Rooney told
ESPN.
"But if I had the opportunity to appoint someone, I'd go all out for Pochettino.
"I think Manchester United - certainly over the last 20, 30 years, since Alex Ferguson - have based themselves on youth players, young players, bringing players through.
"I think [Pochettino] gets the best out of his players whether they are younger or older.
"You see some of the young players he has brought through at Tottenham, at Southampton that have gone on to play for the national team, so I think he ticks every box in terms of his quality as a coach, but also his willingness to give youth a chance."
Solksjaer's arrival at the club in December has seen several players improve their form, including World Cup winner Paul Pogba.
Speaking about Pogba's relationship with Mourinho, Rooney said: "Well I think with Paul, I watched him come through the academy at United and he's got a lot of ability.
"But the difficult time he was going through, if you don't have a good relationship with the coach, then it's difficult to be at your top if you feel like the coach is scrutinising every decision you make, every pass you make.
"Paul Pogba will give the ball away, he'll try passes [and] it won't be the right pass at times, but you have to let him do that because two or three of them passes will create goals, will create opportunities.
"I think he found it difficult under Mourinho. I think a player and a manager with big egos clashed and the outcome was never going to be great."