Baines started in the Three Lions' 5-0 win over San Marino at Wembley on Friday and looks likely to start on Tuesday night in Poland.
Cole, 31, is two games away from reaching 100 caps for England although Stuart is adamant that he will be pushed for his place by the 27-year-old Toffee.
"It was difficult to have a bad game against San Marino but nonetheless Leighton, I thought, was England's best player on the night. I thought he did terrifically well," Stuart told talkSPORT.
"I think he's inching closer and closer to Ashley Cole with every passing week, I really do. His performance level for Everton has been outstanding.
"I think Cole has been absolutely magnificent over the years and for the first time in a long time Ashley will have to be looking over his shoulder now because Leighton is coming up strong."
Baines has 12 England caps to his name.