Labour deputy leader Tom Watson has said the party will “get a grip” on anti-Semitism following the suspension of Ken Livingstone on Thursday.
Labour was considering changes to its rules to send “a clear signal” of its “zero tolerance” towards anti-Semitism.
Mr Livingstone was suspended after claiming Hitler supported Zionism “before he went mad”,
Shadow home secretary Andy Burnham said allegations must be dealt with “much more speedily in the future”.
He told BBC One’s Question Time on Thursday that he did not think the party was anti-Semitic but added: “These allegations, when they are surfacing, have not been dealt with properly and quickly enough”
But Mr Watson told BBC Radio 4′s Today programme Mr Corbyn did “act swiftly” to suspend his ally of 40 years, Mr Livingstone.
He said Mr Livingstone’s comments were “vile” adding: “To link Hitler and Zionism in the way he did must have been designed to create offence.” But he said it was for Labour’s ruling National Executive Committee to decide whether to expel him from the party.
In the meantime he said he and Mr Corbyn had been looking at whether “Labour’s own structures” needed changing “to make sure that we send a very clear signal to people in our party that we will have a zero tolerance approach to anti-Semitism”.
“Do we need to change our rules to explicitly rule out racism and specifically include anti-Semitism in that?,” he said.
“We are going to deal with this. We will take a zero tolerance to anti-Semitism.”
Meanwhile BBC Wales understands that Mr Corbyn has been asked not to join Labour campaigning Wales ahead of the assembly elections following the anti-Semitism row.
Former London mayor Mr Livingstone was suspended by the party, a day after MP Naz Shah was suspended over comments on Facebook.
The row exploded on Thursday after a radio interview that Mr Livingstone did with BBC Radio London, defending Ms Shah, the party’s MP for Bradford West.
In it he said: “When Hitler won his election in 1932 his policy then was that Jews should be moved to Israel. He was supporting Zionism before he went mad and ended up killing six million Jews.”
He was later confronted outside the BBC’s Millbank studios by Labour MP John Mann who accused him of being a “Nazi apologist” in front of TV cameras. Other Labour MPs condemned Mr Livingstone’s comments.
The former London mayor was later suspended by his long-time ally, Mr Corbyn, who said there had been “grave concerns” about the language used.
Mr Corbyn said: “Anybody that thinks this party is not cracking down on anti-Semitism is simply wrong. We have suspended where appropriate, we have investigated all cases. We will not tolerate anti-Semitism in any form whatsoever in the party.”
Labour peer Lord Sugar said Mr Livingstone had “lost the plot” and was “obsessed with Hitler, concentration camps and Jews”.
“They will wait for it to blow over a little bit and they will find some way of him coming back in, slipping back into the Labour fold,” he told BBC Radio 5 Live.
But Jon Lansman, chairman of Momentum, a grassroots network of Corbyn supporters, said that while Mr Livingstone’s comments were “very ill-judged, wrong and offensive” to “call Ken a Nazi apologist is ridiculous”.
He said: “Just as the Tories have an interest in stoking up the flames of whatever crisis is here … largely a crisis of perception .. so do some people in the Labour Party and I’m afraid John Mann is one of them. John Mann is no supporter of Jeremy Corbyn.”
But Rachel Reeves former shadow cabinet member, told BBC Two’s Newsnight there was a “growing problem of anti-Semitism in the Labour Party”.
“Over the last few days and weeks we’ve seen more and more instances of people coming out with views which really have no place in the Labour Party. But also part of the problem is the slow response from the leadership of the party. We do need to see much swifter and more decisive action.”
There was also some criticism of the decision to reprimand John Mann – who was told by Labour’s chief whip that it was “completely inappropriate for Labour Members of Parliament to be involved in very public rows on the television”.
Ilford South Labour MP Mike Gapes said Mr Mann should have been “praised not criticised for his dedicated opposition to anti-Semitism”.
On Wednesday Labour MP Naz Shah was suspended, pending an investigation, over comments she made on Facebook before she became an MP, including a suggestion that Israel should be moved to the United States.
She has apologised but it has prompted claims from senior Labour figures that the party was not doing enough to tackle growing anti-Semitism in its ranks.
Mr Corbyn has denied the party is in crisis, saying: “It’s not a crisis. There’s no crisis. Where there is any racism in the party it will be dealt with, it will be rooted out.”