"He's a very successful manager, nice person! Be louder than you have ever been -- that's absolutely okay. That's how I would say hello to a good old friend," Klopp, hoping to emulate Benitez's success at Liverpool, told a news conference.
"He still lives around here, nobody has to ignore this wonderful moment. Do how you feel!"
Spaniard Benitez spent six years at Liverpool and famously took them to the Champions League title in 2005 when they recovered from 3-0 down against AC Milan in Istanbul to win the final on penalties.
He also took them to the final in 2007, won the FA Cup in 2006 and regularly had Liverpool challenging near the Premier League summit, although he was unable to win it.
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Benitez, who had spells at Inter Milan and Real Madrid after leaving Liverpool, is now in the middle of very different challenge, that of trying to keep Newcastle in the top flight after replacing Steve McClaren.
What looked like a hopeless cause now looks possible, with a victory against Swansea City and a draw with Manchester City giving them a chance of safety with four games left.
"They don't look like a relegation team, but that shows how strong this league is. It's pretty difficult to change direction," Klopp, whose side are five points behind fifth-placed Manchester United with a game in hand, said.
"They are in a good moment but we want to keep the points here, that's why we play on Saturday and so that's what we have to show."
Klopp also offered an update on striker Divock Origi, who was carried off on a stretcher with an ankle injury during Liverpool's 4-0 thrashing of Everton in the Merseyside derby on Wednesday.
He said the Belgium international could still return this season.
"It's an injury, it's serious, but we will try everything, everything to ensure that his season is not over without taking risks for the player -- that's how it is," he said.
"I'm long enough in this business to know we are all different, all individuals, so we don't have to say any general thing. We only have to work with him."
(Reporting by Martyn Herman; Editing by Toby Davis)
