The pair had just pitted their wits in a Premier League thriller which Liverpool edged 3-2 to keep their title hopes alive.
The Guardian report that, after the match, the two managers embraced warmly in a touching moment of respect between the pair away from the cameras.
Liverpool will take their points total to 97 if they beat Wolves at Anfield on the last day – but even that might not be enough.
Manchester City must drop points against either Leicester tonight or Brighton next Sunday – and captain Jordan Henderson is praying for a miracle.
He said: “We’re putting up a great challenge. City are a great team but so are we. We’ve competed right to the very end.
“Whoever gets it will deserve it.We couldn’t have done any more.
“It’s going to the last game and we need to finish on a high at Anfield. We’ve got a game on Tuesday, Barcelona, a big game, and we’ll have to focus on that, but I’m sure the lads will watch the game tonight and, like I say, pray for a miracle.
“Leicester, who are a great team, will give everything, like you saw from Newcastle. I’m sure Brighton will, I’m sure Wolves will.
“Everyone wants to cause an upset, so hopefully there are a couple of twists and turns to come.”
Since Liverpool started their eight-game league winning run on March 10 against Burnley, they have exchanged victories with City. This time they have got their win in first, increasing the pressure on Pep Guardiola‘s side – who have won 19 of their last 20 games, and not lost since January 29.
Liverpool beat Newcastle with a goal from Divock Origi four minutes from time, after Fabinho won a crucial free-kick with what looked like a dive.
On such margins the Anfield charge was kept alive, and Klopp said: “At the very beginning, when I came to Liverpool, I realised this club has to do it the hard way, so we’ll do it the hard way, because luck is not constantly with us.
“We have to really work for it and the boys are ready and the whole club is ready for that. Whatever happens, it will be a very, very successful season.
“We cannot do more – that’s it. For us, the target is winning football games as often as possible, and the boys did much more than I thought possible.”