The move would more than triple Barrett's salary which is believed to be $1 million per season under his current contract with NZ Rugby and the Hurricanes.
Japanese fullback Ayumu Goromaru currently holds the title as World Rugby's most lucrative earner bringing home $2.44m a season with French side Toulon, while former All Black Dan Carter is the highest paid New Zealander at $2.44m with Racing Metro.
Earlier in the week, Barrett made headlines after Crusaders assistant coach Ronan O'Gara stated that the 26-year-old playmaker could surpass Carter as the greatest first-five of all time.
"He fills grounds. He excites people. Everything good about rugby is what Beauden Barrett does and he's an exceptional talent," O'Gara said ahead of the Crusaders' clash against the Hurricanes on Saturday.
"He's probably nearly redefined number 10 play. Dan was obviously and still is the best 10 in the world, overall for decades. But Beauden Barrett is very exciting and could potentially go on to challenge him for that title. So when you speak of him in that class, by God is he up there."
Barret is widely regarded as one of the best players in the game and has picked up back-to-back World Rugby Player of the Year titles.