Victory over Tottenham at the weekend made it three successive Barclays Premier League wins for the first time since May 2011. The result moved them to within two points of Arsenal, in fifth, and seven adrift of fourth-placed Chelsea, although both clubs have a match in hand.
"We said it was always going to be tough but we set ourselves the little task of trying to win as many games as we could in the last 10 and that's three in a row now so we've got a chance," Downing told Press Association Sport.
"We've got some winnable games but that's easier said than done. We have to go and win them but from the way we are playing I think we can beat anyone. The teams in and around that top half are playing each other so obviously some are going to slip up and drop points and the thing for us is to take each game as it comes and see where we finish."
There is a clear feeling there has been progress made under Rodgers, particularly since January and the form guide backs that up. After his first 14 matches in charge (up to the end of November) Liverpool had just 16 points and were 12th in the table.
In the subsequent 15 they have collected 29, bettered only by Manchester United and Spurs - and have moved up to sixth.
Downing expects that to continue throughout the remainder of this campaign and, with new recruits arriving in the summer, into the next.
"I think getting everyone back fit, getting our strongest players out there, has helped with everyone in their proper positions," he added.
"I think everything is coming together. There were signs earlier in the season but it was always going to take time to click - now you are seeing some good signs. There was going to be a learning process this season, building for the future.
"If we can keep this squad together, add a few more and keep the big players, I think we have a hell of a chance."