Montague, who has spent 17 years on Today, will swap roles with another Radio 4 presenter, the World at One's Martha Kearney.
Montague recently criticised the BBC for their "unfair" pay practices in an interview with Radio Times. She said: “I get paid very well for a job I love, but I didn’t know how unfair it was. Also I didn’t realise how professionally damaging it was.
"Mishal [Husain] and I took the uncomfortable step of going public after the pay disclosures, and what that opened up was the scale of the problem. Hopefully, it will lead to change. There is an underlying pay gap that there should not be. Why should somebody be paid less for doing the same work?”
This July, it was revealed that Today's John Humphrys had earned between £600,000 and £649,999 in the year to April 2017. Meanwhile, Montague is understood to have an annual salary of less than £150,000, far below those offered to her other co-hosts, Nick Robinson (£250,000-£299,999) and Mishal Husain (£150,000-£250,000).
Montague, Kearney and Hussain were all among the signatories of an open letter to the BBC's director-general Tony Hall, calling on him to "do the right thing" and "act now" over the gap between male and female employees' pay.
Speculating on the reasons for Montague's departure, the Daily Mail quotes unnamed "insiders" as saying that she felt "a bit elbowed" by the show's other presenters, adding that the World at One would give the 51-year-old "her own canvas".
In her Radio Times interview, Montague said: “The BBC should not be doing things unfairly. We sit in that studio and challenge people over being unfair, so the BBC, of all places, should be fair. In terms of whether something is morally the right thing to do, then the BBC should be way ahead of the curve.”
Montague's comments follow the resignation of her colleague James Harding, head of BBC News, who will quit his post in January 2018 in order to set up a new network with a "distinct approach to the news", aiming for the kind of coverage that the BBC "can’t, and probably shouldn’t do".
The Today programme celebrated its 60th anniversary this year, but the show has come under increased scrutiny in recent months, following the appointment of the Evening Standard's Sarah Sands – a print journalist with no prior experience in radio – as its new editor.
Responding to her decision to introduce new guest presenters to Today, The Telegraph's radio critic Gillian Reynolds wrote: "It sounds as if Sands is filling the programme with her pals".