keep an eye on sth/sb.表示留意某物,照看某物,比如：
Can you keep an eye on my lugguage while I'm away.我走開一下,你能不能幫我照看一下包.
About the book you mentioned,I'll keep an eye on it for you.關(guān)于你要的那本書,我會幫你留意的.
你說的另外兩個表達沒有見過,應(yīng)該是錯誤的.另外有幾個習(xí)語是用的one's eyes,但后面的介詞不是on,而是形容詞open,或者peeled.即：keep one's eyes open/peeled.意思是仔細觀察.