Dalglish tested positive on Wednesday while attending hospital for treatment on gallstones but was discharged on Saturday after showing no symptoms of COVID-19.
The news of Dalglish's test had an emotional impact on the current Liverpool squad according to Klopp, who said the situation was discussed in a group on the WhatsApp messaging platform.
"It was a real shock three days ago when I heard about it first," Klopp told Liverpool's official website.
"The boys were sent a message in our WhatsApp group and everybody was like, 'wow'.
"What you feel in that moment is a massive difference if you know somebody who got the virus, or if you don't know.
"In this moment, it was like, 'wow, one of us has it' and it was really crazy."
Dalglish arrived at Liverpool as a player in 1977 and won six league titles and three European Cups during a spell that saw him become player-manager from 1985 until 1990.
Klopp indicated the closeness between his players and Dalglish, saying: "We all know this terrible disease is causing heartache all over the world, but this was the first time for many of us someone we have such a personal connection to was affected to this extent.
"I had the opportunity to text immediately with one of his daughters and we spoke about it. She was quite, not relaxed, but she was fine and said it looked all well – and two days later we heard he was released from hospital.
"It's good news – very good news – and I hope he is doing well still.
"We all know Kenny and we love him. We just sent him all our thoughts and prayers in that moment, but maybe he didn't need it, which is even better."
The 69-year-old had been admitted for treatment on gallstones on Wednesday and was given a routine coronavirus test, which came back positive.
Dalglish, who was showing no symptoms, was released on Saturday and is now in self-isolation at home.
Writing in The Sunday Post, the former Celtic and Scotland star paid tribute to the NHS staff who looked after him.
"They were absolutely brilliant," he said. "As a nation, we are all very fortunate to have them and I wish them all well as they work tirelessly to help the country through this pandemic."
Dalglish won four Scottish league titles and five domestic cups with Celtic before signing for Liverpool in 1977.
A glittering Anfield career delivered six league titles and three European Cups as player and then player-manager from 1985 until 1990.
Dalglish was in charge during the 1989 Hillsborough disaster, in which 96 Liverpool fans died during a crush in an FA Cup match. He won widespread acclaim for his support of the victims' families.
Dalglish won the Premier League in 1995 as manager of Blackburn Rovers and later had spells in charge of Newcastle United and Celtic before returning to Liverpool for 2011-12, in which they won the EFL Cup and reached the final of the FA Cup, losing to Chelsea.
The news was announced on Friday, in a message from Dalglish's family that said the 69-year-old is asymptomatic.
Scotland great Dalglish had been attending hospital on Wednesday for treatment on an infection when he was tested for COVID-19 and "unexpectedly" was shown to be carrying the virus.
Rush, who partnered Dalglish in Liverpool's attack in the 1980s and is Liverpool's record scorer, wrote on Instagram: "Wishing a speedy recovery to the best...Sir Kenny Dalglish. Get well soon #YNWA"
Gerrard captained Liverpool during Dalglish's second spell in charge at Anfield, which ended in May 2012, and the pair have a close relationship.
Now manager of Rangers, Gerrard wrote: "Get well soon king."
Liverpool's current goalkeeper Alisson sent his well wishes, writing: "Love from Becker family to Sir Kenny Dalglish!!"
Reds past and present, Robbie Keane and James Milner, both sent "Get well soon Sir Kenny" messages, echoed by former England striker Gary Lineker and the Scottish Football Association.
Celtic, the club where Dalglish began his playing career, said: "Sending our love and best wishes to @kennethdalglish following tonight's news. Get well soon, King Kenny."
Newcastle United and Blackburn, who he also managed, also rallied behind Dalglish.
Former Blackburn centre-back Colin Hendry, who helped Dalglish's team to the 1994-95 Premier League title, backed the Scot to return to full health.
Hendry wrote on Twitter: "To one of the biggest influences on my career...my boyhood idol .. my then manager.......King Kenny. Another match you'll win...I'm sure xx"
The Reds issued a statement provided by Dalglish's family, explaining the 69-year-old former Scotland international was in hospital regarding a different matter on Wednesday but was given a COVID-19 test.
"Sir Kenny was admitted to hospital on Wednesday, April 8, for treatment of an infection which required intravenous antibiotics," the statement read.
"In keeping with current procedures, he was subsequently tested for COVID-19 despite having previously displayed no symptoms of the illness. Unexpectedly, the test result was positive but he remains asymptomatic.
"Prior to his admission to hospital, Sir Kenny had chosen to voluntarily self-isolate for longer than the advised period together with his family. He would urge everyone to follow the relevant government and expert guidance in the days and weeks ahead.
"He would like to take this opportunity to thank the brilliant NHS staff, whose dedication, bravery and sacrifice should be the focus of the nation's attention at this extraordinary time.
"He would also ask that they are given the space to do their jobs during what is an extremely challenging time for them and that his own family's privacy is respected.
"He looks forward to being home soon. We will provide further updates as and when it is appropriate."
Dalglish is widely recognised as Liverpool's finest player. He joined the club from Celtic, where he had also enjoyed great success, in 1977, and went on to win six league titles as a player with the Reds, including one as player-manager in the 1985-86 double-winning season.
He also won three European Cups as a player with Liverpool, thriving under bosses Bob Paisley and Joe Fagan before taking the reins himself.
His first spell as manager ended in February 1991 and took in the Hillsborough disaster, when his leadership and his compassion towards the bereaved in the aftermath showed Dalglish at his best off the pitch.
Dalglish led Liverpool to their most recent English league title in the 1989-90 season, before managing Blackburn Rovers to Premier League glory in 1994-95.
He later took charge at Newcastle United, before a second spell as Liverpool manager followed, Dalglish in command at Anfield from January 2011 until May 2012.
The attacker, who lifted four top-flight titles and the European Cup three times during his time at Anfield, won eight caps for the Three Lions and scored six times.
A member of England's Euro 1980 squad, Johnson - nicknamed 'Doc' - was a part of the Liverpool squad that dominated football at home and abroad under Bob Paisley.
Dalglish, who lined up alongside the forward in their 1-0 victory over Real Madrid in the 1981 European Cup Final, honoured his late team-mate with a statement on social media.
"Sad news about The "Doc"," he wrote on Twitter. "David was a really good guy, [a] great team-mate and hugely popular in the dressing room. Our condolences [go out] to all his family."
Another former Reds team-mate, David Fairclough, also paid tribute, adding: "So sad to hear my great friend David Johnson has passed away today. [We] shared so many great moments and memories."
Liverpool themselves also posted a note of condolence, stating: "The thoughts of everyone at the club are with David’s family and friends at this very sad time."
Johnson, who started his career at Merseyside rivals Everton before a move to Ipswich Town, made his England debut in 1975 against Wales, scoring a brace in a 2-2 draw.
Five years later, he won his eighth and final cap in the Three Lions' Euro 1980 opener against Belgium, playing no further part in the tournament as his team fell short in Group 2.
