Suntory, which also owns Orangina Schweppes, is believed to be in advanced talks to buy the brands, after owner GlaxoSmithKline put them on the block earlier this year.
Drugs giant Glaxo concluded Lucozade and Ribena would grow faster under a new owner when it kicked off their sale in April.
A sale to the Japanese drinks giant would mark the latest high-profile British consumer brands to fall into foreign ownership after Branston Pickle was sold to Japan's Mizkan last year and KP Nuts and Hula Hoops were bought by Germany's Intersnack in December.
A deal with Suntory, which bought Orangina in 2009, would circumvent an expected auction.
The brands are said to have attracted considerable interest from private equity groups including CVC Capital Partners and KKR, and were also reportedly in the sights of Irn-Bru owner AG Barr.
The drinks are largely sold in Western markets and earn annual sales of more than £500 million a year.
Lucozade and Ribena, which were owned by Beecham prior to its merger with SmithKline, date back to 1927 and the 1930s respectively.
Glaxo chief executive Sir Andrew Witty has said it would sell the brands if it could recoup "appropriate value".
He said: "We concluded that the tremendous growth potential of these iconic brands, particularly outside the core Western markets, could be better leveraged by companies with existing category presence and infrastructure in these regions."
Lucozade and Ribena form part of Glaxo's consumer healthcare division, which also includes the brands Horlicks and Panadol.
In the three months to the end of June, Lucozade sales grew 4% but Ribena was down 2%.
In July, Glaxo said it hoped to agree their sale by the end of the year.
Glaxo and Suntory declined to comment.