Terms of the deal were not disclosed and no comment was available from the company, which announced the deal in a statement. However, the Herald understands the sale price was around the $1 billion mark.
Advent International is a US$50 billion private equity firm, with investment experience in the payments and software industries.
In the past decade it has invested US$3.8b in 9 payments companies and $3.8b in six software companies around the world, the statement said.
TSG chief executive Steven Holmes said in the statement the partnership with Advent would fund the next phase of its international expansion.
The deal with Advent follows the sale by TSG shareholder Calera Capital of its stake in TSG to Advent. Terms were not disclosed. Calera, a middle-market private equity firm, invested in TSG in 2015.
Reports say during Calera's investment period, TSG grew its annualised payment processing volumes to a value of US$4b and more than 5 million customers.
The deal is a major one for Bolton, a former Brierley Investments and Skellerup chief executive, who appeared on last year's NBR Rich List with an estimated net worth of $165m.
Bolton is also known for his 67 per cent stake in Corporate Cabs and ownership of the 32ha Kamo Wildlife Sanctuary in Northland which he acquired from 'Lion Man' Craig Busch.
UK company office records show TSG Holdings (UK) group revenue last year was A$105.5m, comapred to A$82.7m the previous year.
Results are in Australian dollars because Australia is TSG's biggest earner.
Adjusted EBITDA was A$44.8m, compared to A$32.9m in 2017.
Profit after tax was A$5.8m, up from $3m the previous year. No dividends were paid in the 2018 or 2017 financial years. Net assets in 2018 were A$175.5m (A$167.2m).
TSG was incorporated in 2015 in England and Wales.