United finished a distant third in the Premier League last season, 33 points off champions Liverpool, and Keane reckons Solskjaer cannot afford for that chasm to repeat itself this time around.
"The remit will be different for Ole this year," Keane told Monday Night Football. "I think last year, it was about getting into the top four - I know they got to three semi-finals - but it will be higher this year.
"They will certainly have to get a bit closer to Liverpool and Manchester City. The points total cannot be the same as it has been over the past three or four years.
"Ole will probably have to get his hands on a trophy. I think the honeymoon period is certainly over for Ole, and the pressure starts to mount now, particularly with the performances like that over the last week or so."