Reports have claimed Manchester City turned down an approach from the Merseysiders about a potential swap deal with Carlos Tevez and the £35million England international.
That came on the back of speculation earlier this month - dismissed by Liverpool - that Newcastle were keen to take their former number nine back to the north-east if they could get a cut-price deal.
Liverpool have refused to comment on the story, insisting they never do on transfer speculation, but after Carroll played a part in both goals in Saturday's 2-1 FA Cup win over Manchester United Dalglish was quick to praise the striker, who has had a tough first 12 months for his new club.
"I thought Andy had a fantastic game," said the Scot.
"He works really hard and gave everything he's got for us.
"He has a lot of attributes for us to work on. I am sure that (performance) will do him no harm whatsoever in terms of confidence.
"The individual awards are all well and good but collectively we want to start showing an improvement and winning things and if we continue with that commitment we have a chance."
What a difference a week has made for Liverpool.
Last Saturday Dalglish openly criticised his players for the first time after a woeful 3-1 Barclays Premier League defeat at Bolton.
Since then they came from behind twice to draw with Manchester City and secure a place in the final of the Carling Cup and then dispatched their arch-rivals United out of another knockout competition.
It has certainly lifted confidence levels at the club but with his side still not consistent enough in the league Dalglish will not be getting carried away.
He was, however, delighted with the response he got from his squad.
"We will have to wait and see what this week means for the season but it has got us in a Carling Cup final and the next round of the FA Cup," added the 60-year-old.
"It started badly for us at Bolton and it is fantastic credit to the players, and shows how much pride they take in the club, that they got themselves back on track as quickly as they did.
"They were two enormous games and coming out on the winning side twice - once on aggregate - deserves great credit.
"It has been a really difficult week for us. To play two good teams like City and United is even more of a task - we've had a good week.
"It is much better than it was last Saturday. It is fantastic for us to progress from a very difficult week."
All the pre-match talk had been about the behaviour of players and supporters as Patrice Evra returned to the ground for the first time since October's spat with Luis Suarez eventually resulted in the Uruguay international being banned for eight matches for racial abuse.
But the focus on Evra took a different shape as, with two minutes remaining, it was his error in getting caught out of position which allowed Carroll to flick on Jose Reina's goal-kick to Dirk Kuyt who fired past David de Gea.
Unusually for a match which still has the biggest rivalry in English football there were very few flashpoints on the pitch.
The main talking point before Kuyt's late winner was the performance of United goalkeeper De Gea, starting his first match of 2012, who was culpable for Liverpool's opener when he allowed himself to be crowded out at a corner - where Carroll played a major part - and Daniel Agger headed home.
Park Ji-sung equalised shortly before half-time, though, and it appeared the tie was drifting towards a replay neither side wanted until Kuyt struck only his second goal of the campaign.
The Holland international, whose only other strike came in the Carling Cup against Brighton on September 21, was rewarded for his persistence as despite losing his place in the team he has never been found wanting for determination and work-rate.
"As a professional you just need to keep going," said Kuyt.
"Sometimes things go very well and when things aren't going well you just need to keep working hard and that is what I did.
"I have the belief in myself but with the players around me supporting me we are all in it together.
"If you want to achieve things you have to do it together, it is not a one-man team."