The Radio 2 DJ was first linked to the show in February, but Chiles later threatened to leave the BBC over the news.
However, a One Show source has now told The Guardian: "A deal has been agreed for Chris to do Fridays."
It is unclear how many editions of the show Evans has agreed to host but an insider claimed that producers are discussing whether or not to make Friday night editions more focused on entertainment news.
"There has been a lot of talking about what happens if entertainment is saved for Fridays," the source said. "Does it change the show too much? It's still being discussed but the fact is that Adrian will not be required on Fridays."
It is thought that Chiles will continue to host the programme for four nights a week, although he has not yet signed a new contract with the BBC.
A spokesperson for the BBC said: "While we do not discuss contracts, Adrian is part of our plans for the future of The One Show."
Meanwhile, Chiles's representative explained: "Adrian Chiles is currently talking to the BBC."
> Hollins to present 'One Show' for a week