I was hired at company X on a project for year on contractual basis. At the end of the 12th month, I had a performance review which was great and helpful.

After that they wanted me for another year but we could not agree on conditions of a new contract so I left (unemployed.)

Meanwhile, I was interviewing at "company Y", the company at which I am very very well aware of the salaries since my past colleagues have been there. Company Y offered me "$X" which is lower than what they usually pay to people of my skill set and experience.

Is it a good argument to let them know that "I am well aware of the general salary at your company, so can you kindly raise it"?

If so, what is the reasonable way to argue this? If not,is there any better way to frame it?