The midfielder, who only returned to action earlier this month after being sidelined since March with a groin injury, was forced to miss Saturday's 2-0 victory at West Brom after picking up an ankle infection.
Dalglish was unable to put any timescale on his captain's likely return date, insisting it would be Monday or Tuesday before he could offer an accurate assessment.
Asked whether he would be fit for England's Wembley friendlies with Spain and Sweden on November 12 and 15 respectively, Dalglish replied: "I haven't got a clue.
"Steven has got an infection in his ankle. He was in our minds for the game, it just flared up.
"It's nothing to do with any injuries before, it's an infection. I suppose it's like having an abcess on your mouth.
"We are off Sunday, we will be back in Monday or Tuesday, so we can be more accurate next week than we can be just now."
Dalglish was thrilled with the display of his team, who were also without their other talisman Jamie Carragher due to a calf strain, and he picked out strike pair Andy Carroll and Luis Suarez for particular praise.
The Uruguay international set up Carroll for the Reds' second goal on the stroke of half-time at The Hawthorns after Charlie Adam had converted a controversial ninth-minute penalty.
Dalglish said: "I've ran out of superlatives to describe Luis. He's a fantastic footballer. Within the group there will be people who excel a little bit more than others.
"For us, it seems to be Luis quite often but that's not to say the rest of them are too far behind. There were a lot of excellent performances out there.
"Him and Andy haven't played as a pair that often but I thought they played really well together. We thoroughly deserved to win the game and some of the football we played was fantastic."
Following draws against Manchester United and Norwich, it was a first Premier League victory in three games for Dalglish's team, who are now seven games unbeaten in all competitions and climbed up to fifth - within a point of third-placed Chelsea.
Dalglish added: "We will just keep bouncing on. At the end of the season we will find out where we finish. It's something we've said many times before and it's not served us too badly in the past.
"We will just take one game at a time. The next match is against Swansea and we will put just as much effort into that one as we did this."
Having failed to mastermind a repeat of last year's victory over his former club, Hodgson was left to reflect on the penalty decision and the West Brom head coach admitted he is concerned by what currently appears to warrant a foul.
Referee Lee Mason had seemed ready to let play continue when Suarez tumbled inside the area under a challenge from Jerome Thomas yet, despite the absence of claims from the Reds forward, referee's assistant Gary Beswick flagged for a foul.
Hodgson, who confirmed midfielder Paul Scharner picked up a "slight knee problem" said: "It does worry me a bit that penalties are being given these days for those type of offences.
"I think the referee like the rest of us didn't see anything but the linesman did and the ref decided to take the linesman's advice."
"But I'm not blaming the referee totally for our defeat because we didn't do enough to win the game. I thought we were second best."
Source: AFP