The Reds boss has also delivered positive updates on two other defenders in Trent Alexander-Arnold, who was injured in the lead-up to Euro 2020, and Joel Matip.
The four players are working on their fitness at Liverpool’s pre-season training camp in Austria.
Virgil van Dijk is back in training after missing most of last season with a knee injury
Joe Gomez has also resumed training after missing much of last season with a knee problem
Defensive problems were among the chief reasons why Liverpool were unable to mount a strong Premier League title defence last term.
The loss of Van Dijk was felt particularly strongly, with the Dutchman playing no part in the campaign after suffering a knee injury in the Merseyside derby at Everton in October last year.
Gomez was also ruled out for the season with a knee problem a few weeks later.
Jurgen Klopp expects Van Dijk and Gomez to take a ‘big step forward’ in the next few weeks
Klopp told the club’s website: ‘Virg and Joe will not have double sessions now for the first few days with the team. They have a second session of the day but it’s with the rehab department, physiotherapy – these kinds of things.
‘That’s for these two. (They) look really good. I don’t know exactly when we can make the next steps but it will happen in the next four weeks. For sure, we will get a big step forward.’
Matip, also a centre-back, did not feature again suffering ankle ligament damage in February.
Right-back Alexander-Arnold hit form late in the season to earn a place in England’s summer plans but then sustained a thigh injury in a friendly against Austria last month.
Trent Alexander-Arnold (left) is back in training after missing Euro 2020 through injury
Klopp said: ‘Joel looks good, absolutely, he is the closest probably to normal training and then we will see when we get him started for playing.
‘Trent was a serious enough injury but now not a major one, I would say.
‘He is close but for the moment, probably the next five or six days he has a special programme – from time to time with the team, from time to time by himself. But then in five or six or seven days he should be in normal team training.’
Van Dijk and Gomez could only watch from the side lines last season as the Reds struggled and laboured before eventually finishing third in the Premier League.
England defender Gomez revealed how he and Van Dijk helped support each other whilst they were both out injured.
Gomez (left) and Van Dijk (right) helped each cope as they fought to return from injury
‘There were so many moments, right at the beginning, when we were at home on the sofa and couldn’t move, and we were FaceTiming and speaking about things. It’s someone to relate to, to vent to at times when you’re frustrated, so it was massive.’ The England defender told Liverpool’s official website.
‘I think we’re both thankful to have had each other; it was cruel circumstances, but it definitely made things easier.’
‘That’s where it’s nice to work alongside each other – we had different injuries, but we’ve been in it together all the way. There’ll be days where we have to do something more suitable for us, but to do team training today after such a long time means a lot. I think speaking for us both, we loved every minute of it.
‘We’re all working towards one goal to try and play our part come the start of the season. Everyone is on their own journeys, but having the people around you to work with in such a good environment and such good facilities, it makes it a lot easier.’