Is 察颜观色 or 察言观色 the correct term to use? I have found citations from ancient texts backing both interpretations on the Internet while searching for the correct form of the term:


  清·唐甄《潜书·食难》：“吾老矣，岂能复俯首于他人之宇下，察颜观色，以求无拂于人，吾不能也。


However, 


  《论语·颜渊》：“夫达也者，质直而好义，察言而观色，虑以下人。” 


Which of these is/are correct?
Both are correct. According to http://www.zdic.net,


  察言观色
  
  察：详审。观察别人的说话或脸色。多指揣摸别人的心意。


察：详审(scrutinize)。言：言语(speech)。观：看(look)。色：脸上表现的神情(expression on face)。
So it means dive sb.'s thoughts from what he says and the expression on his face.


  察颜观色
  
  观察别人的脸色，以揣摩其心意。


Note it's a little different that here's only watching expression on sb.'s face, no listening the speech.

And 漫说“察言观色”与“察颜观色”,


  “察颜观色”与“察言观色”各有其特定的语境可用，而且都有较远的语源可查，更有权威工具书为据，现代文学中又有许多经典著作的用例佐证，当让这两个成语‚各司其职‛，为传承汉语精确性服务。