The midfielder was troubled by a groin problem last season and the club announced last week that he would miss the start of the forthcoming campaign because of an infection that required hospital treatment.
At a press conference to introduce summer signings Stewart Downing, Charlie Adam, Jordan Henderson and Alexander Doni, Dalglish said: ''Steven won't be too far away. All the tests that have come back are far better than what was anticipated so we're delighted with that for him.
''He's had some really positive news and we'll just wait for the progress to continue. But the date that has been mentioned hasn't been changed.''
While Gerrard will certainly be missed, Dalglish has concentrated on strengthening his midfield in his summer recruitment, with Downing, Henderson and Adam estimated to have cost more than £40 million between them. The Scot has been impressed with their impact so far, and said: ''Everybody you bring in you hope is going to improve the quality you already have here.
''That's what we're hoping from all of them, just to be themselves and go about their work as well as they did at their previous clubs. They're pleased that they've come here and we're delighted to have them.''
The Merseysiders' upturn in form towards the end of last season, coupled with their financial outlay, means expectations among Liverpool supporters will once again be sky high for the new Premier League season.
Pre-season has been mixed, with the Reds conceding three goals in each of their five games, including defeats against Hull and Galatasaray, but Dalglish is confident things will be different once he fields his full-strength side.
He added: ''It could have been better but we've had a lot of people coming in at various times. Everybody that has been with us has got a reasonable amount of time on the pitch. As we get closer to the Sunderland game we'll pare it down a little bit.''