Clarke, Jose Mourinho's former No 2 at Chelsea, has been out of work since he left West Ham last June, following Gianfranco Zola's sacking.
But with Dalglish looking for someone with experience to come work alongside him and Sammy Lee, he phoned Clarke on Sunday evening to broach the subject and the 47-year-old did not need to be asked twice.
Come in No 2: Steve Clarke lost his job at West Ham in June
He has signed a deal until the end of the season but Damien Comolli, Liverpool's director of football strategy, has indicated there is scope for those terms to be extended.
'I was sat at home doing nothing and now I've got the chance of working with a football great in Kenny Dalglish and also working for one of the world's leading football clubs,' Clarke said.
'I am really looking forward to working with Kenny and the rest of the staff to help bring good times back to the club.'
Roy Hodgson suffered during his time at Liverpool from his decision not to look elsewhere after his attempt to lure former defender Sami Hyypia back to the club was rejected.
Clarke, who won two Premier League titles as coach at Chelsea, already knows Joe Cole and Glen Johnson from their spell together at Stamford Bridge but he introduced himself to the rest of Liverpool's squad at Melwood on Monday when he took training with Dalglish and Lee.
'This was something I couldn't turn down,' he said. 'I am grateful for everybody who has given me such a warm welcome, especially all those at Melwood.'
Straight to work: Steve Clarke (left) watches on in training with Kenny Dalglish
Sportsmail's Jamie Redknapp said on Monday that one of the first things Dalglish should do is appoint current vice-captain Jamie Carragher to the backroom staff and it is not hard to envisage the 32-year-old occupying some kind of role in the future.
Even though he is currently out with a dislocated shoulder, Carragher travelled with Liverpool's squad to Old Trafford for the FA Cup tie on Sunday and sat on the bench behind Dalglish.
For the moment, however, Dalglish wants Carragher as a player and he said: 'Carra's doing his (coaching) badges and I'm sure that in the long term he wants to be involved in football.
'But at this moment in time he is on the way back from a dislocated shoulder. He was at the match there and it was great to see him.
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'There will be discussions after this but right at this moment he is just coming back That's not to dismiss anything else that will happen in the future.'
Dalglish's return has undoubtedly lifted the atmosphere in Liverpool's squad and Holland forward Dirk Kuyt was the first player to speak about the Scot's impact.
'Kenny Dalglish is a legend and a very important man for this club,' said Kuyt. 'I think it says everything about him as a man that he is here to help us and put us back where we belong.
'It is very sad that Roy has gone because you can never just blame one man. We must all take the blame and perform better. The most important thing is to make Liverpool a winning team again.
'It's disappointing to lose a manager like that but a very big plus that we can bring in someone like Kenny. He is a very experienced man and someone who can have a positive impact on the squad.'
Kenny Dalglish steps into the red.and admits he would love to walk on and on at Liverpool Time for a new chapter! Kenny Dalglish's Liverpool return has Anfield faithful buzzingKop that! King Kenny back at Anfield as Clarke is unveiled as his No 2Steve Clarke named Liverpool's new No 2 under Kenny DalglishLIVERPOOL FC