According to the Daily Mirror, the Strictly Come Dancing finalist is understood to have been formally offered the job in a deal thought to be worth £200,000.
The 32-year old is expected to accept the BBC's offer later this week.
He will join the Welsh presenter Alex Jones on the One Show sofa on Monday to Thursday evenings, while Chris Evans will continue to co-host the extended Friday evening show.
Baker has been co-presenting the topical magazine-style show since Manford left last month after admitting that he exchanged sexual messages online with female fans.
Earlier this week, Baker said he was keen to take up the One Show full time, but ruled out stepping down from his Countryfile duties.
The former Blue Peter presenter will continue to present BBC1's Countryfile programme on Sunday nights.
If Baker accepts the offer, he will be the One Show's fourth male co-presenter in eight months, following Adrian Chiles, Manford and Evans.