Square closed up 7.89 percent at $9.30 following the news, after rising as much as 14 percent during the trading session.
The stock held gains even after Visa noted that some early reports on its regulatory filing were erroneous.
"Visa has not bought or sold Square shares since its original investment," the payments giant said in a statement to AFP.
Visa added that "various news outlets have incorrectly reported that Visa is currently a top five percent shareholder of Square Inc., currently owning 9.99 percent of Square's Class A common stock."
The statement noted that Visa holds "approximately one percent of Square's fully diluted common equity as of December 31, 2015."
While Visa holds private shares from its 2011 investment before Square's public offering, it has an option to convert the stake to Class A or public shares which would represent around 10 percent of that share class, according to the filing.
Square's rally began early in the session and held despite the initial confusion from some of the reports.
Mark Palmer at BTIG Research, in a note to clients early in the day, said the news "was particularly notable insofar as it served as a vote of confidence in the company's business model, its brand, and its ability to continue to grow both domestically and internationally."
The analyst also said the Visa stake "should serve as a reminder of Square's attractiveness as an acquisition target."
The San Francisco firm led by Dorsey, who is chief executive of both Twitter and Square, launched its initial public offering in November, raising $243 million.
Square, which started in 2009 by providing financial transaction software for smartphones or tablets along with free "dongles" that read magnetic strips on payment cards, was among prominent "unicorns" with valuations of at least $1 billion in startup-rich Silicon Valley.
But its value slipped from a level of $6 billion during its private fundraising phase to around $4 billion ahead of its IPO.