The majority of home fans left the stadium before the final whistle and those who remained booed loudly as the team recorded only two shots on target.
West Ham are 15th on 27 points, three places above the league's final relegation spot. Everton (16th), Nottingham Forest (17th) and Bournemouth (18th) are also on 27 points but have played a game more than Moyes's side and have an inferior goal difference.
Asked whether he felt the board could consider making a managerial change, Moyes said: "I've got no doubt they are behind me. I'm confident in the way we work and what they think."
The former Manchester United boss took responsibility for the team's defeat and acknowledged the mounting pressure on his role.
"I'm a big boy and have left other jobs in the past. If this one happens, I would have to go with that," he added.
"I really like my job here. I like the people I work for and have enjoyed my time greatly. I'm hoping there are a couple of big days ahead in the not too distant future."
