Mr Kroenke has gradually increased his holding in Arsenal
Stan Kroenke has bought 25 more shares in Arsenal and is within 17 shares of the threshold that would force him to make a takeover bid.
US billionaire Mr Kroenke, a director at the club, now holds a 29.9% stake.
Takeover guidelines in the UK state that if a shareholder's stake in a company reaches 30%, he or she has to offer to buy the entire stockholding.
Mr Kroenke has gradually increased his stake in the club this year but has declined to comment on his intentions.
Mr Kroenke already owns the NBA's Denver Nuggets basketball side and the Colorado Rapids MLS football team.
He paid £212,500 for the 25 shares.
Arsenal's second-biggest shareholder is Russian tycoon Alisher Usmanov, who owns about 26% of the club's shares.