European reports say Beziers also want former All Blacks midfielder Ma'a Nonu and Welsh sharpshooter Dan Biggar.
The club is in the middle of a deal to be bought by wealthy investors from the United Arab Emirates.
Beziers are currently battling in the highly competitive French second division but the once-proud club – they have 11 French titles to their name – are desperate to be a major force in Europe again and are prepared to go on a spending spree to achieve that.
The club said they held a “constructive” meeting in Paris with the new investors where the current shareholders were reassured about the authenticity of the UAE based bid and their “financial capacity”.
The buying spree would see the club need to work within the salary cap of €11.3 million (NZ$19.7m), with an extra €200,000 (NZ$350,000) allowed per French international.
Barrett is currently excelling with the Blues in his shift of allegiance in Super Rugby Aotearoa.
Nonu was playing for San Diego in Major League Rugby in the United States before it was put on hold by the coronavirus pandemic.
Bezier's' plan sounds similar to that successfully undertaken by Toulon who signed major New Zealand talent like Tana Umaga and Sonny Bill Williams to lift them out of the second division and become a Top 14 powerhouse and one of the most dominant clubs in France in recent years.