Wijnaldum has become a key member of Klopp's squad in recent seasons, making 37 Premier League appearances last term as Liverpool ended their 30-year title wait.
The Netherlands international was also influential during the Reds' Champions League success in 2019, though has now entered the final year of his contract at Anfield.
Media reports have linked the player with Barca, where his former national team boss Ronald Koeman is in charge, but Klopp was unwilling to discuss the matter in great detail on Friday.
Speaking in his pre-match press conference ahead of the Community Shield against Arsenal, the Liverpool boss said: "That's an open contract situation, otherwise we would know.
"It's all between us and the player, these kind of things. It's all good at the moment, all fine. Gini played very well in pre-season and has obviously had a couple of good seasons for us.
"That's all I can say about that."
The reigning champions may have lined up reinforcements in midfield, however, if rumors of a move for Thiago Alcantara are to be believed.
The Spaniard helped Bayern Munich achieve the treble but has made clear his desire to move on for a new challenge, though the financial impact of the coronavirus pandemic could restrict the amount of business Liverpool do during this transfer window.
"I don't know, I don't know what will happen. There's a long time to go until October 6 [when the window closes]," Klopp replied when asked about potential new signings in general.
"I think the philosophy of this club is pretty clear. We will see how much we can, or how much we want to spend, all these kind of things."
The emergence of Rhian Brewster, who scored twice in the pre-season friendly against Salzburg this week, could negate the need to add further firepower upfront.
Alternatively, the 20-year-old forward could be sent out on loan again, as was the case in the previous season when he had a temporary spell in the Championship with Swansea City.
"We have to see what will happen in the next few weeks or so. We want to use Rhian but want to help him as well, that's why we have to make decisions," Klopp said about Brewster.
"He looks really promising. He can obviously score goals, that's good. All the rest? We will see. He's still a young boy, especially with his position, a number nine upfront. We have to see where we can give him experience, here or somewhere else.
"We haven't decided yet - and why should we? There's absolutely no reason to rush anything. He's our boy and he's here."