Fawaz, Abdulaziz and Omar al-Hasawi issued a joint statement saying they were in negotiations with the estate of Nigel Doughty to buy the Championship club.
"We are currently completing our due diligence in preparation for the exciting challenges ahead of us," said a statement on the club's website.
A statement from Forest's board said Doughty's estate had granted the Hawasi family "an exclusive period of negotiation".
Multi-millionaire Doughty died in February aged 54, having invested almost £100 million in a failed attempt to revive the club's fortunes.
Forest are one of only eight clubs to have retained the European Cup, securing their second crown in 1980, but they have fallen on hard times and have not played in the Premier League since being relegated in 1999.
The City Ground side ended last season in 19th place in the second tier.
"We wish to bring the club back to where it belongs in the top division. Together we can bring the glory days back," tweeted Fawaz Al-Hasawi on Friday.
His family's main business interests are in property, hotels and air conditioning, while Fawaz recently quit as chairman of Kuwait's Qadsia, which has won the past four domestic championships.
The Hasawis are rarely mentioned in descriptions of Kuwait's wealthiest and most powerful families and so Forest's potential new owners are unlikely to rival the lavish spending of Abu Dhabi's Sheikh Mansour at Premier League champions Manchester City.
On June 25, a company called NFFC Group Holdings Ltd was registered with Companies House. Omar al-Hasawi is a director of this firm, according to the Director Check website.