Azerbaijan does not have many options, in an Ideal world Azerbaijan would acquire F-16s, as its two closest defense partners(Turkey and Pakistan) also use F-16s, so there is extensive familiarity and Turkish origin F-16 munitions and sensors could be ported over.
Unfortunately, The Armenian lobby will not allow the US congress to approve any sales of US aircraft, nor will it allow any aircraft with US engines or subcomponents to be sold to Azerbaijan. So the F-16, Gripen, Korean T-50 are all off the table. The French will obviously not sell Azerbaijan a Rafale, so thats off the table. The UK might want to sell the Eurofighter, but Germany will in al likelihood be opposed and block. The Tejas India would refuse(not that Azerbaijan would buy either way), plus it has a US Engine. Other various aircraft producers also either use US engines or their fighters are still in development, so not yet viable.
This leaves only 2 options Russia and China. While Azerbaijan operated the Mig-29 and has significant familiarity with Russian Aircraft due to soviet part, this is not exactly the ideal solution b/c of proximity and reliance on Russia, particularly with the regional politics, Russia would be hesitant to sell anything that would change the balance of power between Armenia and Azerbaijan, as well as potentially being a liability if any tension with Azerbaijan happened. Azerbaijan would also be reliant on Russia for support of any platform.
So the only logical answer left is CHINA. As of right now, China has two options for aircraft available for export. the JF-17 Block 3 and the J-10C.
Both are good choices with their own unique upsides. If money were not an issue, between these 2, the J-10C would be the perfect choice, as its in the same class as the F-16s that Turkey flies, and many munitions can be ported over.
The JF-17 Block 3, while not as capable as the J-10C, provides incredible capabilities when you factor in the price point. While it doesn't have the payload capacity of the J-10C, it has many similar sensors and weapons packages.