He looks odds-on to leave the post in the summer.
Large sections of the home support at Upton Park jeered each of Allardyce's three substitutions in the second half before a chorus of boos greeted the full-time whistle.
Mile Jedinak had put Palace in front with a penalty just before the hour mark and West Ham rarely looked like scoring an equaliser as they struggled to create chances and conceded possession cheaply on a number of occasions.
It is not the first time the Hammers players have experienced a hostile atmosphere from their own supporters this season after the team were booed off last month despite beating Hull 2-1.
Sam Allardyce: "I always have sympathy with the fans' frustrations.
"I'm as frustrated as they are and I don't want to see my team lose a game of football, particularly at home.
"At the end of the day though it's difficult when you are playing a team that is very good at what they do.
"Palace have hardly conceded more than one goal in any of their last 10 or 11 games so we knew we had to score the first goal and try to win the game 1-0.
"If we got more than that then fine but our finishing failed us and perhaps the referee failed us too."
West Ham are seven points clear of the relegation places.