(CNN) -- Gov. David Paterson had no intention of appointing Caroline Kennedy to fill the Senate seat vacated by Hillary Clinton, a source close to the New York governor told CNN Thursday. 

Caroline Kennedy has withdrawn her name from consideration for Hillary Clinton's Senate seat. 

The source told CNN that Paterson did not think Kennedy was "ready for prime time," citing her efforts, at times awkward, to try to win the appointment. 

"She clearly has no policy experience and couldn't handle the pressure," said the source, who spoke on the condition of anonymity because of the sensitivity of the situation. "Why would he pick her given how badly she handled herself in recent weeks?" 

Kennedy, 51, cited personal reasons for her decision not to continue pursuing the Senate seat. She informed the media at midnight, as Wednesday turned to Thursday. 

Paterson is charged with naming a replacement for Clinton, who resigned her seat to become the secretary of state in President Obama's administration. Paterson will appoint someone to hold the seat until a new election is held in 2010. 

A Kennedy ally, though, denied Kennedy had any indication Paterson was leaning against choosing her to fill out Clinton's term. 

And another Kennedy confidante said Kennedy allies are getting frustrated about what they perceive as the governor's insiders slighting her. 

Paterson, who is expected to name his choice Friday, has been coy about who he will pick to replace Clinton. Kennedy had been very public in expressing her interest in the seat by meeting with state and community leaders throughout the state. 
Who is the Govenor?
Gov. David Paterson 