"I'm looking forward to carrying on working with the team over at ITV."
Peter Davey, commissioning editor of ITV Entertainment, said: "We thank Caroline and Olly for all their hard work on The X Factor and the contribution that they've made over the past few years.
"We look forward to working with Caroline again soon and wish Olly the very best as he returns to concentrate on his music career."
Flack had previously hosted The Xtra Factor for three years from 2011 until 2013.
Last year, she and Murs were chosen as new co-presenters to replace long-time X Factor host Dermot O'Leary.
Pop star Murs said he had chosen not to return to the ITV show later this year in order to dedicate more time to his music.
In an interview published in The Sun on Monday, he said: "This was an incredibly hard decision to make and one I didn't take lightly as I've really enjoyed co-hosting The X Factor.
"It's been one big adventure and I loved every minute of it. But my heart belongs to music and I want to concentrate 100% on making my next album and getting back to performing."
Richard Holloway, managing director of Thames TV and interim chief executive of FremantleMedia UK, said: "Caroline and Olly were a brilliant double act on The X Factor, from their early days on Xtra Factor to last year at the helm of the main show.
"I'd like to thank Caroline and Olly for their contribution to the show and wish them all the best."
On Friday, it emerged that X Factor judge Nick Grimshaw had also told boss Simon Cowell he was quitting as a judge.
Despite the exodus from the judging and presenting team, the programme makers insisted there would be another series.
Thames TV, which makes the talent show, said in a statement: "Details for the new series of The X Factor are still being discussed and will be announced in due course."
Earlier this month, broadcaster ITV said it intends to continue with the programme for at least two more years.
A spokesman said: "The X Factor and Britain's Got Talent contracts come up for renewal after the latest series air later this year, and we fully intend to continue with both shows into 2017 and beyond."
There has been no announcement about whether judges Cheryl Fernandez-Versini and Rita Ora will return in late 2016 alongside Cowell.
Speculation about the future of the Saturday night singing contest has grown since the latest series saw disappointing viewing figures.
Audience numbers averaged 8.2 million, with 8.4 million tuning in to watch the December final - significantly down on the programme's heyday.
The X Factor withstood another blow at the end of the 2015 series in December when winner Louisa Johnson reached only number nine in the charts at Christmas.
This was the lowest entry for any winner over the show's 12-series history.
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