Netherlands international Van Dijk had surgery after suffering knee ligament damage in October.
Gomez was injured on England duty in November and had an operation to repair a tendon in his left knee.
Joe is not running, Virgil is already running but this is a real tough one, said Klopp.
The Euros are taking place between 11 June and 11 July, having been delayed by a year because of the coronavirus pandemic.
Talking about Van Dijk and Gomez being involved at the tournament, Klopp said:
It is not my decision but, to be honest, the information I have at the moment is that it will be unlikely.
It’s not that I don’t let them go, it is because of the extent of the injuries.
We all hope they will be ready to start pre-season with us, that’s what we all hope. With Joel [Matip] it is pretty much the same.
These are really serious injuries and it is now not about talking about which competition they play.
“I am always open for positive surprises and they all of a sudden turn up in team training – but no-one told me that.
Matip was ruled out for the season after suffering an ankle ligament injury at the end of January and Klopp said the centre-back is still in what looks like ski boots.
We will be completely over the moon when they are ready to train with us for the start of pre-season but the Euros, I didn’t think about it, added Klopp.
When do you have to be back to be ready to play in the Euros? The week before? The day the Euros start? I don’t know.
I understand maybe the need of that but we cannot make this decision and we will not. They will be fit when they are fit.