Solskjaer took over from Jose Mourinho in December after the Portuguese was sacked following poor results and contributing to low morale in the dressing room .
The Norwegian took the managerial reigns on an interim basis at the time, but a brilliant turnaround in form has seen many call for his permanent appointment.
Under Solskjaer, the club have won 10 of their last 12 fixtures with the only loss coming in the Champions League on Tuesday against a brilliant Paris-Saint Germain side.
No one at the club foresaw such an impressive run of results when he was hired in December, with the Red Devils initially just viewing him as a stop-gap between now and the summer.
However, the club now face the dilemma of appointing him on a longer basis as he continues to guide them to more victories.
He has brought back a winning mentality at the club and helped to boost the atmosphere, something which has arguably been missing at times since Sir Alex Ferguson left in 2013.
Now, Neville believes the 45-year-old should be given the job on a full-time basis if the results continue and insists hiring someone else could cause further unrest.
“If Manchester United continue in this way until the end of the season and get into the top four then the risk with not giving Solskjaer the job would be huge,” Neville told Sky Sports.
“”I still think you don’t need to announce it today”Gary Neville
“If you get to October and they have spent £30m or £40m to get a new coaching set-up into the club and they are fifth, all hell will break loose. People will say: why did you not give Ole the job when he was doing it well?
“I still think you don’t need to announce it today because as soon as you do it’s ‘who are you going to sign and who are you going to get rid of’. At the moment it’s still a very good atmosphere, but once the job comes then the questions would start to change.”
Marcus Rashford and Paul Pogba have been a revelation under Solskjaer, besides the latter’s red card on Tuesday night.
The English striker has bagged six goals for the club since his arrival while Pogba has scored eight himself, including a brace against Fulham on Saturday.