George Carey also said Labour party members should do more to tackle anti-semitism in their ranks.
And he warned that if they failed to do so, it could cost them victory at the next general election.
Labour has been hit by a number of anti-semitism controversies under Mr Corbyn's leadership.
Last August, he said he was "sorry for the hurt that has been caused to many Jewish people" and admitted the party had been too slow to deal with disciplinary cases against Labour members accused of anti-semitism.
But in an interview with i24NEWS to mark Holocaust Remembrance Day, Lord Carey - who was Archbishop of Canterbury from 1991 until 2002 - cast doubt on Mr Corbyn's commitment to tackling the problem.
He said: "The weakness of his statements can give the impression that he is, deep-down, somebody who doesn’t like Jewish people."
The leading cleric added: "I would encourage socialists and those in the Labour party to pay attention to what they say about racism and about the influence of Jews within their own party.
"I want to repeat again that a great party such as the Labour party should not put up with this. It should find ways of uniting itself around one common core cause and that is to make this nation great again."
Lord Carey went on: "I think this might give Jeremy Corbyn as leader an opportunity to speak out really clearly and to silence those who question where he stands on this issue.
"He has not done it so far, but there’s going to be plenty of opportunity between now and an election to do so. Otherwise it could cost him the election."
Elsewhere in the interview, Lord Carey also said the Christian Church could do more to tackle the rise in antisemitism.
He said: "Christians are to blame as well. We send parties of Christians from our churches to Israel, they are there for one week or ten days. They see Palestinians, and they come back with a prejudice view of what’s going on, and it’s a one sided view.
"What I think we need to do more is to educate Christians who go on these pilgrimages to see this as not the total story. What they don’t realise is what is happening to the Jewish communities in their own land, who are side-lined and in a way persecuted by Palestinians. You’ve got to get a much more balanced view of what’s going on."