The report cites sources close to the deal who say Sun CEO Scott McNealy has been holed up with his staff for the last few days, hammering out an offer.
One source is quoted as saying, "There will be an offer on the table soon." Another undisclosed source close to Apple says the struggling Cupertino, California, firm is expected to accept it.
Although details are sketchy, the report states the deal could take the form of a $4 billion stock swap, which would translate to $33 for each Apple share.
Although Apple chief executive Michael Spindler is set to face the music at the company's annual shareholders meeting in Cupertino today, it is unlikely that a takeover offer will be presented at that time. Sources cited in the Wall Street Journal report say it will probably take a few more days for the deal to be in place.