Nuno is expected to be in place before Monday's Premier League match against Everton, having said he has had positive talks with the West Ham board.
Potter was dismissed on Saturday morning after just eight months in charge, with the club 19th place in the table.
The Hammers took only three points from their opening five Premier League games this season under the Englishman.
The east London club said they believed "a change is necessary in order to help improve the team's position in the Premier League as soon as possible".
They added: "Results and performances over the course of the second half of last season and the start of the 2025-26 season have not matched expectations."
Potter, 50, had been preparing for Monday's game and fulfilled his media duties on Friday, saying he believed he had the board's backing after "positive talks".
After last Saturday's home defeat by Crystal Palace, BBC Sport reported the club were considering managerial alternatives, and discussions with Nuno have accelerated over the past 72 hours.
Nuno, 51, was sacked after 21 months in charge at Forest - and just three games into this season.
Last term the Portuguese guided the club to seventh in the Premier League - their highest finish since 1994-95 - as they qualified for Europe for the first time in three decades.
Potter took charge of 25 matches after being appointed in January - winning just six times - and the Hammers finished last season in 14th place in the league.
His loss rate of 52% is the highest of any permanent West Ham manager in the Premier League era.
"The board would like to thank Graham and his coaching staff for their hard work during their time with the Hammers and wish them every success for the future," the club added.
"The process of appointing a replacement is under way. The club will be making no further comment at this time."
