The Labour leader said the deaths of Grace O'Malley-Kumar and Barnaby Webber, both 19, and Ian Coates, 65, who were stabbed by Valdo Calocane last June were "absolutely awful".
The police have been accused of having "blood on their hands" after a series of failures meant the killer was able to carry out the attacks.
Barnaby's mother, Emma, said "true justice has not been served" after Calocane was handed a hospital order and not jailed. And she told Nottinghamshire Police assistant chief Rob Griffin: "If you had just done your job properly, there's a very good chance my beautiful boy would be alive today."
But Downing Street said it was “important” that as “the first action the relevant agencies look back and ensure that all the proper processes were followed and that reasonable steps that could have been taken were taken, to ensure that where there are lessons to be learned we do so.”
The Attorney General is also considering whether judges should review the sentence to decide if it is unduly lenient.
The 32-year-old has paranoid schizophrenia and his pleas of manslaughter by diminished responsibility were accepted earlier this week.
But the families of those killed reacted angrily to the sentencing and accused prosecutors of a "fait accompli" in not pursuing a murder verdict.
Sir Keir, who in a previous job was the top prosecutor for England and Wales, told ITV's This Morning programme: "As far as the sentence is concerned, obviously there are mental health issues in this particular case, and the Attorney General has got the power to review it and I think that probably makes sense and have it double checked by the Court of Appeal.
"But I think alongside the sentence, I am very worried by what appear to be a number of points at which action could have been taken that would have prevented this happening.
"The family are saying that needs to be an inquiry into that. I think they're right about that. I think somebody outside of this independent needs to look at exactly what happened, what were the points of which there could have been an intervention and why it didn't happen. That is the least that these families are owed."
Rishi Sunak has so far resisted ordering an inquiry.
A Number 10 spokeswoman said: "We believe that it's important that as the first action, that the relevant agencies look back and ensure that all the proper processes were followed and that reasonable steps that could have been taken were taken, to ensure that where there are lessons to be learned we do so.
"That is the first thing that needs to happen."
Judge Mr Justice Turner said Calocane would "very probably" be detained in a high-security hospital for the rest of his life as he sentenced him for the "atrocious" killings, as well as the attempted murder of three others.
Speaking on the steps outside Nottingham Crown Court after the sentencing, Mrs Webber said: "We were presented with a fait accompli that the decision had been made to accept manslaughter charges.
"At no point during the previous five-and-a-half-months were we given any indication that this could conclude in anything other than murder.
"We trusted in our system, foolishly as it turns out."