Downing, with the England squad in Sofia for Friday's Euro 2012 qualifier against Bulgaria, has made an impressive contribution to Liverpool's unbeaten start after his £20m move from Aston Villa.
He said: "We've certainly got the quality to challenge. It's going to be difficult because Manchester City have strengthened and have got a hell of a squad while Manchester Utd have started really well
"We've got a hell of a chance of doing it but we have got to pick up points and stay in and around that pack to have a chance of doing that."
Downing added: "No disrespect to the other clubs, but at Liverpool they are expecting to win games. You can see the expectation, spending big and having a pretty big revamp. We've started very well. We don't want to look too far ahead because you could slip up.
"You can sense the expectations. It was a difficult couple of seasons with stuff going on with the owners and that but now there is a buzz around the place.
"The new owners have come in and shown their intentions by spending good money on players and revamping it so expectation goes up and Liverpool fans always want to see success. If the team keeps playing the way it is at the minute then we have got a hell of a chance of succeeding."
And Downing has settled quickly at Liverpool under Dalglish, admitting the fact that the Scot was willing to pay £20m to sign him was a major lift to his confidence.
"I feel more confident as a player because the team at Liverpool is playing quite well," said Downing. "I'm personally happy with my form. I just want to continue that and hopefully if I get a chance with England I can play well and stay in the team.
"The fact that a man like Kenny Dalglish is willing to spend £20m on me does my confidence the world of good. He's put a lot of faith in me paying that amount of money. It does give me a lot of confidence. As soon as I went to Liverpool I felt quite settled, great club, great team and it's great to be part of it."