Airbus claims that it takes 1 month for the final assembly of an A320 and from manufacturing of first part to completed aircraft it takes 1 year. This is probably because Airbus has various factories all across Europe to build the planes.
After the fuselage is brought over by train, Boeing has claimed that it only takes 9 days to assemble the aircraft. They are currently planning to make 42 737s per month from February.
From a pretty long time, Airbus has had an edge over Boeing in terms of A320 per month, currently it seems to be approximately 45-60.
Boeing 737 in 9 days after receiving fuselage
Airbus A320 per month figures (Aug 23)
Time taken for A320 manufacturing
PS: I see now that the question was asking about pre-2019 production rates, from some reasearch I've done it can be seen that Airbus was still a bit ahead of Boeing in 2019 but Boeing was certainly more competitive than it is now.