Sky News understands that Bestway is one of two remaining bidders for IDH Group, which runs about 600 practices across the UK.
Both Bestway and Bridgepoint, the private equity firm which used to own Oasis, another big dentistry business, are said to have indicated that they would be prepared to offer more than £700m for IDH.
A successful offer from Bestway would take it into another area of the healthcare industry, and represent the latest in a series of audacious takeovers by the Pervez family-owned group.
In 2014, it acquired the Co-operative Group's pharmacy business for £620m, and has successfully built it into the biggest independent chain in Britain - and the largest behind Boots and Lloyd's.
More recently, it acquired Costcutter to expand its wholesale operations.
Bestway now ranks as the UK's 17th-largest privately owned company, according to its website.
Sources said that Carlyle and Palamon, IDH's existing owners, were likely to decide on a preferred bidder in the coming weeks.
It is unclear whether the current shareholders will see a meaningful return because of the level of debt within IDH.
Bestway and Carlyle declined to comment.