Martin O'Neill's side looked out of contention after losing 7-1 at Chelsea last month and much of the focus has been on Tottenham and Manchester City in the scrap for fourth place in the Premier League.
But Villa have responded to that mauling at Stamford Bridge by taking 13 points from the last 15 to move level on points with Tottenham, although they have played a game more.
They take on Roberto Mancini's side at Eastlands on Saturday before ending the campaign with a home game against Blackburn.
Downing said: "Saturday is a big game. With the way everyone has been going on, they're expecting Tottenham or Man City to get that fourth place and nobody has said anything about us.
"I think it suits us being regarded as underdogs because we'll just carry on winning games and picking up points and trying to creep in through that back door.
"Manchester City have obviously spent a lot of money on big players and it's been the same with Tottenham. They've had a good go as well for fourth place.
"But we've a good team and we've played consistently well. We've had a good season, got to a cup final and cup semi-final, but we'd like to finish if off by getting that fourth spot.
"Now we've got two big games left and we have to try and finish the job off."
Downing insists Villa will not be fazed by the weekend clash with City, whose first-choice keeper Shay Given will miss the remainder of the season through injury.
He said: "We play quite well on the counter-attack when we're away from home. It's going to be a massive game but one we look forward to.
"It's unfortunate for Shay getting that injury but it's a chance for another keeper to come in.
"They've got good players all around, they've got a good squad - but we're not afraid of them."