"I don't know where to start," said Keane. "I knew they weren't in a great place but I am shocked and saddened by how bad they are. You can lose a game of football but everything about United was lacklustre.
"No quality. A lack of desire. A lack of leaders. A lack of characters. It is a long way back and it is scary just how far they have fallen."
Since their stirring victory over Paris Saint-Germain in last season's Champions League, the result which helped Solskjaer secure the manager's job permanently, United have lost 10 of 19 games, winning just five.
Mourinho, who was sacked in December and replaced by Solskjaer, said: "We were bad last season but I do not see any improvements, even with three new players [Harry Maguire, Aaron Wan-Bissaka and Daniel James].
"I have to say they are three new players with a positive impact. I like these three players and they are bringing some good quality but the team, as a team, I don't like at all. I am not surprised by the result and I don't think Ole can take any positives from the game."
It was United's second successive away defeat by West Ham, who struck through Andriy Yarmolenko and Aaron Cresswell and had control of the match throughout.
Solskjaer rued key moments of the game which slipped through United's grasp as he insisted his side showed the desire he is looking for in his team.
"It is a good group to work with," Solskjaer said. "They are determined. Sometimes along the road, we are going to hit some bumps, some highs and some lows. The group is out there determined, they showed the desire we want to get out, it was a good atmosphere in the dressing room. They have got the focus right.
"They want to do what we are trying to do. We are working in training. There are key moments that can tip these games in our favour and you make maybe a bad decision on a pass, but the attitude and desire is there. That is not the problem. It is the quality in our decision-making, sometimes in the execution."
Gary Neville, another former United captain, said Solskjaer will need four transfer windows to complete the construction of a playing squad he described as "depleted".
United are now hoping the injured Anthony Martial and teenage sensation Mason Greenwood, who was ill yesterday, will recover in time to face Arsenal next week.
"The squad is depleted and the away stand towards the end was depleted," said Neville. "I can't blame them. The coaches that have come in have been given money over the last seven, eight years but they have been let go when they need another year.
"Ole needs another three or four windows to get this squad up to the standard and a couple of hundred million pounds to do so. You can bring through young players, but you need that spine."
