The Whites dropped into the bottom three at the weekend after losing out to Arsenal at the Emirates while Everton collected a vital three points against Leicester City at the King Power Stadium.
Jesse Marsch's men are level on points with the Clarets, who have picked up 10 points from their last five games under interim manager Mike Jackson, but an inferior goal difference has sunk them into the relegation zone.
Leeds entertain World and European champions Chelsea at Elland Road on Wednesday evening knowing that a point will see them switch places with Burnley, who travel to the Tottenham Hotspur Stadium on Sunday.
With three fixtures remaining this term, Woodgate, who made 100 starts for Leeds in the Premiership, and featured for the club in the UEFA Cup and Champions League, said: "I hope they [Leeds United] do stay up! It’s hard to judge. Everton have been brilliant the last few games, beating Chelsea at home and beating Leicester away and the victory against Man United as well.
"I was looking at their away form and thinking they hadn’t won since Brighton on August 28th – six points away from home since before Leicester, and I think they’ll be okay now as they’ve got a couple of home games and I think they’ll pick up six points from those. It’s going to be down to Burnley and Leeds.
"I watched Jesse Marsch’s first game against Leicester, and I know they got beat in that game, but you could see what they were trying to do and they were very good.
"They’ve been playing very well, and I thought they were clear of the danger, but have been dragged back in by the form of Burnley and Everton. But it’s out of Burnley and Leeds!"
Former England international Woodgate was speaking to BritishGambler.co.uk.