Reigning Masters champion Rahm last week joined LIV in a deal worth hundreds of millions of dollars in a huge coup for the upstart tour.
LIV had already recruited major winners Brooks Koepka, Phil Mickelson, Dustin Johnson and Cameron Smith, but Rahm was arguably the biggest name yet to leave the PGA Tour.
Speaking from Saudi Arabia to the BBC's World Business Report, the Australian Norman -- the public face of LIV -- said there would be "more apples falling from the tree".
"To have Jon on board is critically important to our next steps into the future and what we want to do," Norman said.
He added: "It will create a domino effect, there will be more apples falling from the tree, there's no question about it because LIV continues to grow and develop."
Spain's Rahm had previously ruled out going to LIV, which held its inaugural season in 2022 and triggered a civil war in golf.
He is reportedly set to earn between $300 million and $600 million.
"Since Jon signed, less than a week ago now, I know my phone is blowing up," Norman added.
"I know we probably have eight to 12 players who are very, very keen to sit down and talk to us about coming on board."
Saudi Arabia's foray into sport has triggered accusations that it is "sportswashing" its human rights record.
Norman, a two-time major winner, said that Saudi Arabia "really cares" about golf.
"If you see the way they're embracing the game of golf, they're investing in the game of golf in their country for their people into the future," he said.
Rahm's move came as the PGA Tour and the Saudi backers of LIV attempt to finalise the details of their bombshell merger announced in June.
A deadline of December 31 has been set to agree a framework for the new PGA-LIV joint venture.
