Tesla is shifting its entire standard range EVs to LFP battery chemistry globally (energy storage systems too in the future), and the entry-level Model 3/Model Y produced in China are already equipped with CATL's LFP cells (you can see an example pack here).
A diversification to engage BYD sounds reasonable, but it's not officially confirmed.
Moreover, it's not known whether Tesla would use BYD's batteries produced in China, or a new production line would be launched in the U.S. or even in Europe. As far as we know, BYD's plant in Lancaster, California is only for electric buses, trucks and battery systems (not the cells).
Update June 8, 2022: CATL contacted us and explained that in the latest CTP 3.0 version announced in March 2022, CATL's LFP battery can reach 160 Wh/kg (see CATL's announcement from March 2022).
Not only that. BYD's solution is very affordable, estimated at €55 ($64) per kWh. That would explain why BYD is now so competitive and is ramping up production and sales so quickly. PushEVs listed in August a few numbers.
A 60 kWh battery at less than $4,000 does not sound bad. But let's wait for confirmation whether Tesla is really working with BYD.