We were seeing a similar noise in our application.
We were able to narrow it down systematically by turning off portions of the codec and seeing if the noise persisted. We found that it was coming from the ADC block, and determined that it was bleed through from the MIC input to the ADC.
Since our design does not utilize the mic input of the SGTL5000, we left it unconnected. Once we identified that the noise was coming in through the codec, we needed to determine if it was something internal to the codec. We grounded the line-in and saw no difference, but when we grounded the mic-in our unwanted tone went away. We then routed a 1kHz tone to the line-in and found that the known input signal was amplified as expected when routed through the line in, but when we configured the codec to connect the mic input to the ADC we saw the same 1kHz tone that we were injecting on the linein (but the mic input was not connected to anything, it just had the left line-in pin routed nearby). It had an amplitude that was nearly equivalent to that coming out of our source. Once we connected a 1uF cap to ground on the mic input pins, we saw our random tones go away.