Michael Carrick has led Manchester United up to third in the Premier League No Premier League side has taken more points than Manchester United since Michael Carrick took over as interim manager, but he still need to see the job through if he is to be considered the number one candidate for the full-time vacancy. Manchester United were seventh in the table when Ruben Amorim was dismissed in January, but they have made up an 11-point deficit on Aston Villa under Carrick to rise up to third. Carrick's former United and England teammates Wayne Rooney and Gary Neville have been understandably impressed by the job he has done so far - but both feel he still has a way to go to fully convince them he is the right man for the job longer term. Article continues below Manchester United should keep manager search going: Gary Neville and Wayne Rooney Rooney, Carrick and Neville played together at Manchester United (Image credit: Getty Images) Speaking on Stick to Football, Rooney said: "I think if they get Champions League football, [Carrick] should get the job. "He puts himself right up there if they're looking for new managers - he definitely puts himself in pole position, I think." Will Spurs Be Relegated? Rooney Backs Carrick & Roy’s Cheltenham Tips | Stick to Football EP 120 - YouTube Watch On Neville agrees that United should continue to keep their options open for the time being and see what unfolds over the rest of the season. But Neville also seems a bit less convinced of whether Carrick should be seen as a viable long-term option given his limited managerial CV and United's prior experience of giving a successful caretaker the job longer term. Get the FourFourTwo Newsletter The best features, fun and footballing quizzes, straight to your inbox every week. Contact me with news and offers from other Future brands Receive email from us on behalf of our trusted partners or sponsors He said: "Manchester United have to pursue another manager right now, [based] around the fact [that] you don't know how the results are going to go. "But little things like Thomas Tuchel re-signing with England does start to take away the options for Manchester United and they don't harm Michael's chances of getting the job. Neville later coached Rooney and Carrick as part of the England set-up (Image credit: Getty Images) "If he gets United into the Champions League there will be a lot of fever around that he should be appointed. "I'm not against him being appointed - I love him to bits - but I think Manchester United should go for the best class of manager available. "They've had young and inexperienced managers in the last two picks. Ole Gunnar Solskjaer has come in before as an ex-player and that didn't work in the end. "So I feel removing as much risk as possible is the right option."