The Red Devils boast a 100 per cent record under the Norwegian coach since Solskjaer took over the reins from Jose Mourinho after a 3-1 loss to Liverpool FC earlier this month.
Manchester United were 5-1 winners against Cardiff City in Wales in Solskjaer’s opening game in charge to boost morale at the Old Trafford outfit.
The 20-time English champions followed up that landslide win with a 3-1 victory over Huddersfield Town thanks to goals from Paul Pogba and Nemanja Matic.
Manchester United have moved to within eight points of fourth-placed Chelsea FC in the Premier League table in the race to secure the final Champions League qualification spot.
The Red Devils will take on Bournemouth at Old Trafford in pursuit of a third successive Premier League victory.
Eddie Howe’s side have lost four of their last six Premier League games and suffered a resounding 5-0 defeat by Tottenham Hotspur at Wembley on Boxing Day.
Former Liverpool FC defender Lawrenson believes the Red Devils will make it three wins out of three when Solskjaer’s improving side take on Bournemouth at Old Trafford.
“Manchester United are on a roll with two wins out of two under interim boss Ole Gunnar Solskjaer but I still think it is going to be very difficult for them to make the top four,” Lawrenson told BBC Sport.
“United have improved because they have got so much quality in their squad and so many of their players had a point to prove, but how long does that last for?”
He added: “I do think they will beat Bournemouth, who are on a terrible run of eight defeats from their past 10 games, but there are tougher tests to come – such as their trip to Wembley to play Spurs on 13 January.
“The Cherries’ form has fallen away since the start of November, when they conceded a last-gasp winner against United – it has all gone horribly wrong for them since then.”
Manchester United are unbeaten in their last six Premier League meetings with Bournemouth since losing to the Cherries for the first time in December 2015.
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