The $29bn (A$39bn; £21bn) deal is set to be Australia’s biggest-ever buyout.
The offer is a more than 30% premium to Afterpay’s stock market closing price on Friday.
The agreement will create an instalments payment giant as the industry sees significant growth.
“Square and Afterpay have a shared purpose. We built our business to make the financial system more fair, accessible, and inclusive, and Afterpay has built a trusted brand aligned with those principles,” Square co-founder and chief executive Jack Dorsey said in a statement.
Afterpay has been seen as a key indicator of the prospects for the niche no-credit-checks online payments industry that jumped in popularity last year as more users, especially young people, opted to pay in instalments for everyday items during the coronavirus pandemic.
The agreement means that Afterpay will be able to expand more quickly in America. Its latest annual figures showed sales in the US nearly tripled for the period to $8.15bn.
At the same time Square announced second quarter earnings that showed gross profit rose 91% to $1.14bn compared to the same period a year ago.
Afterpay shares were trading more than 20% higher in Monday’s trade in Sydney.