United kept faith with Solskjaer earlier this season after a slow start to the Premier League season and their Champions League group stage exit.
The Red Devils are now seen as the only team with any hope of thwarting Manchester City in the title race.
Champions Liverpool are 13 points behind leaders City having played one match more with Jurgen Klopp conceding after a 3-1 loss to Leicester on Saturday that they are out of the race.
United, meanwhile, can trim the gap to City to five points by beating West Brom today (2pm kick-off) although they will have played one match more than their neighbours.
Solskjaer’s charges are also into the quarter-finals of the FA Cup, where they will face third-placed Leicester, and resume their Europa League campaign this month against Real Sociedad.
United fell at the semi-final hurdle in the Carabao Cup, FA Cup and Europa League last season and finished third in the league but 33 points off top spot.
And Neville thinks that Solskjaer needs his first trophy, having been in charge since December 2018, to come this season or next term if he is to avoid being sacked by United.
“He knows the Manchester United standard,” he told The Webby and O’Neil YouTube channel. “Have we improved since he took over? Are we better to watch than the day he took over? Yes.
“Let’s go back to that last six months of Jose [Mourinho]. It was going pear-shaped. Let’s go back to that team and we had those players on the pitch. We had [Alexis] Sanchez. Sanchez is a great player, but you were looking at them and thinking, ‘I don’t like them’.
“And then now, I really like the players on the pitch. I like them. You might say, ‘all of them just need to go up a level,’ or ‘we need two or three more’.
“But the manager that has come in has improved the team – we’re better than we were. The positions in the league are getting back to a standard where they were in that season with Jose when they finished second.
“We need to win a trophy. The lads need to win a trophy. But we’re better and we like them more. It’s progression.
“It can’t stop cos we’re nowhere near where we need to be. But there is progression.
“And people say, ‘Ole’s this, Ole’s that’. He’s tough. He knows where he needs to get to. We’re better than we were, and in these next 18 months, he’s got to get there. He’s got to get there.
And if he doesn’t get there, he’ll have had three and a half years to get there, that’s enough time… So, things are better, but it’s whether they can go that final notch. Hopefully they will.”
Solskjaer meanwhile feels his side have proven this season that they can go on to win trophies, although a first title since 2012-13 may be beyond them with City on a 16-match winning run across all competitions.
Ahead of their trip today to the Hawthorns to face the Baggies, United have only won two of their last six, losing at home to Sheffield United while drawing to Liverpool, Arsenal and Everton.
Man Utd chief draws up Lingard transfer plan after West Ham start
Solskjaer snubbing Ferguson as Man Utd try to keep Man City race alive
Man Utd boss Ole Gunnar Solskjaer teases Erling Haaland transfer swoop
“It’s an important spell,” the Norwegian coach said. “It’s time now that every game you tick one off and there’s less and less possibility to catch up on points on the teams in front of you.
“That’s what we want to do. That’s what we’re here to do. Sunday’s a big one for us in that respect to get back on winning ways.
“The two points dropped against Everton was disappointing when it’s the last kick of the game.
“But we bounced back really well in the cup against West Ham. We’re through, we’re in the quarters and that was a good step in the right direction again, another clean sheet.”
United have a crunch clash with City to look forward to next month on March 6 but before then face Sociedad over two legs in Europe as well as Newcastle and Chelsea in the league.