Parker is believed to be West Ham's top choice to replace Nuno and could be appointed in time for Wednesday's match against Cardiff after opening talks with the club.
The former England midfielder spent four seasons as a player with West Ham and led Bournemouth and Burnley to promotion from the Championship as a manager.
Parker, 46, also had spells in charge at Fulham and Club Brugge.
West Ham are looking for their fourth permanent boss since David Moyes left in May 2024.
Julen Lopetegui was sacked after just six months in charge and Graham Potter was axed after only eight months in charge.
Nuno, 52, oversaw the Hammers' relegation from the Premier League last season but stayed with the east London club in a bid to lead them back into the top flight.
Many viewed the Hammers as favourites to win automatic promotion, but they are currently eighth in the second-tier table, six points behind leaders Middlesbrough, after 13 games.
West Ham had lost their past two league games against Southampton and West Brom.
Nuno, who previously managed Wolves, Tottenham and Nottingham Forest, was appointed in September last year.
"West Ham United can confirm that men's first-team head coach Nuno Espirito Santo has left his position with immediate effect," the club said in a statement.
"Following a thorough review of the start to the EFL Championship season, the board of directors believe that a change is necessary at this time in order to ensure that the club's objectives and expectations remain on track."