The Scot, who turns 51 on Friday, lasted just 10 months as Sir Alex Ferguson’s successor having guided the club towards their lowest finish in Premier League history, which will also result in an absence from the Champions League for a minimum of 18 months.
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"He certainly had a hard start last summer when he took the job," Keane told ITV.
"The chief executive [Ed Woodward] didn’t get the deals done that he wanted, there was a lot of negativity and then they had a slow start and had no momentum.
"I think it is a shame. He should have been given more time. Some of the players at Manchester United should be really ashamed of themselves, they have really let him down.
"For David Moyes to be seventh in the league is disappointing and he should take responsibility for that but you still need people around you to help you in every way you can. I don’t think he’s had that support, particularly form the players. I definitely think he deserved some more time."
Sir Alex has faced criticism over his decision to choose Moyes as his successor but Keane believes the 72-year-old should not be blamed.
"I wouldn’t necessarily say Moyes was his pick," said Keane.
"He certainly put his name forward but obviously the club had to make the decision so I don’t think you can be critical of Sir Alex Ferguson."