The report claims that Phillips, who is known for taking shows like Strictly Come Dancing global, may not think Evans is the man to do that with Top Gear.
"Matt is more well-known and popular with audiences, so Chris will take a back seat like he did towards the end of the series."
The final episode of the current series will air on BBC 2 this Sunday.
The report claims that the talks are set for later in July, after the Beeb has time to analyse worldwide viewing ratings.
These talks will likely be delayed as Phillips does not start her new role until September 5.
The fourth episode of the revamped BBC Two motoring show was broadcast last Sunday, with overnight ratings slumping to a new record low of just 2.3 million.
This was down from the previous week's total of 2.4 million, which matched the lowest in the programme's history, and a further fall from 2.8 million for episode two and 4.1 million for episode one.
Top Gear's future is reportedly in jeopardy and not just for low ratings; Chris' co-host Matt LeBlanc has reportedly threatened to quit the new-look programme unless the red-haired funnyman is sacked.
Reports released on Sunday suggested that the former Friends actor is becoming increasingly frustrated with how rude his co-presenter is with the staff on set and has told producers that they can no longer work together.
