Rapper Bobby Vylan, of rap punk duo Bob Vylan, on Saturday led crowds on the festival's West Holts Stage in chants of "Free, free Palestine" and "Death, death to the IDF (Israel Defence Forces)".
A member of Belfast rap trio Kneecap suggested fans "start a riot" at his bandmate's forthcoming court appearance related to a terrorism charge.
Responding to the chants from Bob Vylan, Prime Minister Keir Starmer said there was "no excuse for this kind of appalling hate speech".
"I said that Kneecap should not be given a platform and that goes for any other performers making threats or inciting violence.
"The BBC needs to explain how these scenes came to be broadcast."
A member of Kneecap said "f*** Keir Starmer" during their performance after the prime minister called for the band not to play at the festival.
Avon and Somerset Police said video evidence from the performances would be assessed by officers to determine whether any offences may have been committed that would require a criminal investigation.
Glastonbury organiser Emily Eavis said Bob Vylan's chants "very much crossed a line".
"We are urgently reminding everyone involved in the production of the festival that there is no place at Glastonbury for antisemitism, hate speech or incitement to violence," she said in Instagram.
Shadow home secretary Chris Philp said Bob Vylan was "inciting violence and hatred" and should be arrested and prosecuted.
"By broadcasting his vile hatred, the BBC appear to have also broken the law," he said.
"I call on the Police to urgently investigate and prosecute the BBC as well for broadcasting this. Our national broadcaster should not be transmitting hateful material designed to incite violence and conflict," he posted on X.
Health Secretary Wes Streeting called it a "pretty shameless publicity stunt" and said the BBC and Glastonbury have "questions to answer about how we saw such a spectacle on our screens".
Conservative leader Kemi Badenoch called the scenes "grotesque".
"Glorifying violence against Jews isn't edgy. The West is playing with fire if we allow this sort of behaviour to go unchecked," she wrote on X.
Culture Secretary Lisa Nandy has spoken to the BBC director general about Bob Vylan's performance, a government spokesperson said.
The BBC said it showed a warning during the performance and that viewers would not be able to access it on demand.
A spokesperson for the broadcaster said: "Some of the comments made during Bob Vylan's set were deeply offensive.
"During this live stream on iPlayer, which reflected what was happening on stage, a warning was issued on screen about the very strong and discriminatory language.
"We have no plans to make the performance available on demand."
The Israeli embassy said it was "deeply disturbed by the inflammatory and hateful rhetoric expressed on stage at the Glastonbury Festival".
The Campaign Against Antisemitism said it would be formally complaining to the BBC over its "outrageous decision" to broadcast Bob Vylan.
"Our national broadcaster must apologise for its dissemination of this extremist vitriol, and those responsible must be removed from their positions," a spokesperson said.
with PA
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