McEachran is on a season-long loan at the Riverside, but the European champions have a recall option if they wished to use it in the New Year.
But the England Under-21 midfielder wants to continue his impressive stint with Boro, who are pushing for promotion.
"I've been at Chelsea since the age of seven and I've grown up there. It's always been my dream to play regular first-team football there," he said.
"But at present, I'd like to stay here for the rest of the season. We've got a genuine chance of going up and I want to be part of it.
"There's a long way to go this season and a lot could happen but we have a really good manager here and I like the way he wants to play.
"The standard of the Championship has impressed me and I'm getting a lot out of it. We're playing two games a week and it's really intensive, but it's improving me as a player."