Looked at the trouble shooting guide on Applicance Parts Pros website which said the "clicking" was likely due to a faulty "Run Capacitor". I replaced the "Run Capacitor" but found that the clicking sound continued to occur.
I then ordered a Start Relay assembly (the assembly that Run Capacitor plugs into) which is Whirlpool part number WPW10194431. After replacing the Start Relay assembly the clicking sound not longer occurred.
In summary, the solution to eliminate the refrigerator "clicking" sound was to replace both the Start Rely (WPW10194431) and the Run Capacitor (WPW10662129).