Aguero, 31, completed his 12th hat-trick in the English top-flight as defending champions City thrashed Aston Villa 6-1 on Sunday afternoon.
In scoring three goals, Aguero also surpassed Arsenal legend Thierry Henry as the top overseas scorer in Premier League history and moved level with Chelsea hero Frank Lampard.
Aguero has developed into one of City’s greatest players since signing from Atletico Madrid in 2011 and has scored 249 goals in 359 appearances overall.
‘Congratulations to Sergio Aguero for breaking my record of Premier League hat-tricks,’ Shearer said in a column for The Sun.
‘He’s a world-class goalscorer and arguably the best overseas player we’ve ever seen here.
‘At least I can enjoy my record of 260 Prem goals for a little while longer!’
Argentina international Aguero, meanwhile, has set his sights on Shearer’s all-time record.
‘I will try,’ he said after Manchester City’s emphatic victory at Villa Park. ‘He has too many goals so I don’t know, but I will try.’
— Sergio Kun Aguero (@aguerosergiokun) January 12, 2020
Aguero added: ‘I am so happy for the record but thank you to my team-mates because they help me.
‘I am so happy. I want to keep scoring more goals but it all depends on my team-mates.’
After thrashing Aston Villa, Manchester City moved back above Leicester City into second and closed the gap on leaders Liverpool to 14 points, although the Reds have a game in hand.