There have been periods of poor form and intense pressure since the Norwegian succeeded Jose Mourinho, initially on a temporary basis, in December 2018.
But the Old Trafford giants kept faith in the 1999 treble hero and have been rewarded with improving performances, with United heading into the midweek fixtures top of the Premier League.
The Red Devils finished third in Solskjaer’s first full season after goal difference saw them edge ahead of Chelsea, who sacked all-time top scorer Lampard on Monday following a rocky recent run.
“It’s not really for me to comment too much on that one, of course,” Solskjaer said. “One thing I know for sure is that Frank doesn’t want any of us to feel sorry for him.
“I know his character and mentality and I’m sure he’ll bounce back and have a very good managerial career.
“They just went through a little patchy period for a short spell – it’s not long ago they were top of the league.
“It’s the first dip of form they really had under Frank so a little bit surprised but it’s not my job to comment on what’s happening at Chelsea.”
But there are clear similarities given the way Solskjaer and Lampard returned to the clubs they shone for as players, albeit United have shown far more patience when the chips were down.
“Well, I don’t know,” he said when asked if sometimes patience was a virtue. “I can just comment on our club.
“We knew we were going to have some rough and patchy periods and they stood strong.
“Of course I’ve had good backing from the club and the leadership have been very steady, which I feel very grateful for because we’ve been through two or three bad spells of form, of course.
“But sometimes we don’t know what’s happening behind the scenes – you probably don’t know what’s happening behind the scenes here either – and it’s not for me to keep talking about our communication but it’s always sad when someone loses their job.”
Solskjaer was the bookmakers’ favourite to be the next manager to leave just a matter of weeks ago, yet now he has United top of the pile at this stage for the first time since Sir Alex Ferguson retired.
The Red Devils are preparing for Wednesday’s clash with rock-bottom Sheffield United buoyed by Sunday’s thrilling 3-2 FA Cup win against rivals Liverpool, leading to talk of a new deal rather than a new manager.
“We haven’t had any discussions yet,” said Solskjaer, whose current contract expires in 2022. “For me, (I’ll) just keep on doing my job as well as I can.”
Solskjaer leads his side into battle against the Blades with his squad in rude health, with Marcus Rashford given the green light after limping off with a knee complaint towards the end against Liverpool.
“Marcus is available,” he added at his pre-match press conference on Tuesday afternoon. “He trained this morning so available for selection. I think he tweaked his knee a little bit but he should be fine. Trained fully this morning so we’re looking strong, which is a good place to be in.”
Rashford scored against Liverpool after providing a superb assist for Mason Greenwood, who is showing signs again of the levels he reached during an outstanding breakthrough campaign.
“We think Mason is going be very important in the second half of the season,” Solskjaer said.
“But it’s not just going happen by itself because you played a good game and scored. He scored against West Ham a while back as well and you expected him to kick on.
“I think Mason has taken steps, he’s grown, he’s more mature and he’s been training with Edinson Cavani for half a season.
“He’s learning good habits and whoever doesn’t learn from a professional like him or Bruno (Fernandes) or the players we’ve brought in, that’s a pity.
“I’m very confident that Mason will chip in with good performances, goals and assists.”