It depends on which specific model from Airbus or Boeing you're looking at, but there are a few major players (and some joint ventures between them) in the business of manufacturing jet engines for airliners.
Nowadays, most manufacturers let airlines choose between engines from two, or even three, different manufacturers for each model family they offer. Within a given "series" of engine, there will be several variations with differing amounts of thrust.
The "big three" who have at least a part of every engine on a Boeing or Airbus airliner are:
General Electric
CF6 series used in Airbus A300, A310 and A330 and Boeing 747 and 767
GE90 series used in Boeing 777
GEnx series used in Boeing 787 and Boeing 747-8i Pratt & Whitney
JT8D series used in early Boeing 737
JT9D series used in Airbus A300 and A310 and Boeing 747 and 767
PW2000 series used in Boeing 757
PW4000 series used in Airbus A300, A310 and A330 and Boeing 747, 767 and 777
PW6000 series used in Airbus A318 Rolls-Royce
RB200 series used in Boeing 747 and 757
Trent 500 series used in later Airbus A340
Trent 700 series used in Airbus A330
Trent 800 series used in Boeing 777
Trent 970 series used in Airbus A350 and A380
Trent 1000 series used in Boeing 787
And the big joint ventures are: CFM International (joint venture of General Electric and Safran Aircraft Engines)
CFM56 series used in Airbus A318/A319/A320/A321, early A340, and later Boeing 737 Engine Alliance (joint venture of General Electric and Pratt & Whitney)
GP7000 series used in Airbus A380 International Aero Engines (joint venture of Pratt & Whitney, Rolls-Royce, Japanese Aero Engine Corporation and MTU)
IAE V2500 series used in the Airbus A319/A320/A321
Just as a piece of trivia - Pratt & Whitney was founded in 1925 and is a subsidiary of United Technologies Corporation. UTC is one of the fragments of the United Aircraft and Transport Corporation, which from 1929 to 1934 owned not only Pratt & Whitney, but also Boeing, and United Airlines!
... 
/https://www.discuss.com.hk/attachments/day_240418/20240418_0d436206ecaa2bfcd6feMEaoGLslvYRH.jpeg)
/https://www.discuss.com.hk/attachments/day_240418/20240418_dc77f2f2211dc0594f17cRGRRV3SVV31.jpeg)
/https://www.discuss.com.hk/attachments/day_240418/20240418_85fcbc4561458f62adbcBUVNa8UfAPXk.jpeg)
/https://www.discuss.com.hk/attachments/day_240418/20240418_7ebb5f2a21898db6d3c7l4CTd734k72S.png)
/https://www.discuss.com.hk/attachments/day_240418/20240418_792ccd9a662ec45eb120VbjNnd2qI3dK.jpeg)
/https://www.discuss.com.hk/attachments/day_240418/20240418_3bb5af66a58191f0ffe6sheHWxgd5uBV.jpeg)
/https://www.discuss.com.hk/attachments/day_240418/20240418_761df8395a384a1300edEyDE11yHX2Np.jpeg)
