Reid made the announcement at his news conference following the Eagles' 38-33 loss to Dallas on Sunday night.
Vick is out with a concussion and Reid said on Monday that Vick had still not recovered from the injury.
Reid says he talked with Vick about the demotion, and Vick was "positive about it" and "on board," but said the decision to start Nick Foles was his.
"Each week he's come in and he's kind of been the replacement guy," Reid said of Foles. "Now he is the starter and we'll see how he does with that on his plate and see how he handles it. "
Reid went on to say Vick would most likely be second string when he gets healthy, saying he wants Vick to "get healthy" and have a long career here.
Foles was 22 of 34 for 251 yards and one touchdown against the Cowboys. He has thrown for 793 yards, two touchdowns and three interceptions and has a passer rating of 73.3. Vick struggled this season, committing 14 turnovers in nine games.
Earlier in the day, the Eagles announced the firing of defensive line coach Jim Washburn.
Taking Washburn's place will be Tommy Brasher, who previously served as the Eagles defensive line coach in 1985 and again from 1999-2005.
Talking about Washburn's firing, Reid said it was not a move to save his job. Rather, he said, it was something he had been "pondering and working through" for some time, saying it didn't work out the way he wanted it to.
In a statement released by the Eagles on Monday morning, head coach Andy Reid is quoted as saying, "Jim is a fine football coach and we appreciate the efforts he gave to this team over the past two years… However, I determined that it was in the team's best interest that we move in a different direction in terms of trying to maximize the production of that position group. We look forward to having Tommy Brasher back on board to work with the defensive line."
The Eagles hired Washburn before last season, expecting his wide-nine defensive line alignment to produce plenty of pressure on opposing quarterbacks. The scheme helped the Eagles tie for the league lead with 50 sacks last season but has resulted in only 20 so far this year.
Washburn's firing comes a day after the Eagles dropped to a 3-9 record.
Last week, the Eagles released defensive end Jason Babin, who excelled in Washburn's scheme while in Tennessee.