A regulatory filing published on Wednesday showed Bestway had added to the 3.45% stake it declared last Friday - a holding it described as an investment and not a step towards making a takeover offer for Sainsbury's.
However, Bestway had said it may look to make further market purchases of Sainsbury's shares, subject to availability and price.
Shares in Sainsbury's were up 0.5% in midday trading, extending gains over the last month to 21%.
A spokesperson for Sainsbury's declined to comment beyond Friday's statement that the group would engage with Bestway in line with its normal interactions with shareholders.
A spokesperson for Bestway also declined to comment beyond restating its position that the stake is for "investment purposes".
There has been speculation that Bestway wants to use the stake as leverage to get Sainsbury's to collaborate in wholesale areas or pharmacy retail.
Earlier this month, Lloyds Pharmacy said it was closing its 237 counters in Sainsbury's stores.
Some analysts point out, however, that Bestway, which is owned by the Pervez, Choudrey and Sheikh families, could have pursued a pharmacy collaboration without becoming a major investor.
The Qatar Investment Authority (QIA) is Sainsbury's' largest shareholder with 14.3%, while the second-largest is Vesa Equity Investment, the vehicle of Czech billionaire Daniel Kretinsky, which has 10%, according to Refinitiv data.
Neither has a seat on Sainsbury's board.
Both declined to comment on Bestway's investment.