The report of the completion of the ICAO fact-finding investigation confirms that not attempt to contact the aircraft was made:
No attempt was made by the USSR to contact the crew of KE 007 by radio on the
distress frequency 121.5 MHz or on any other VHF or HF frequency.
The report also contains a summary of an interview with the pilot of the interceptor plane. He said that he did not have time to contact KE007 because he would have lost connection to ground command:
The ICAO team was unable to meet with the SU-15 interceptor pilot in February 1993,
but was provided with articles published by Izvestia in January 1991 containing extensive interviews with
him. The contents of the articles were confirmed as authentic by representatives of the Russian
Federation. [...]
He also reported that
he did not try to establish radio contact with the aircraft because he would not have had the time to do
so, he would have had to tune to that frequency and in so doing he would have lost contact with his
ground command.
(emphasis mine)
The pilot also reported that he did not know it was a passenger aircraft:
The interceptor pilot stated that 1983 was a difficult year for Soviet interceptor pilots in
the Far East region as there had been numerous intrusions into Soviet airspace by military aircraft of the
United States. [...]
At that stage he could more clearly see the aircraft, but could not identify its type
as Soviet pilots did not "study" foreign civilian aircraft. The flashing lights (rotating beacon) of the
aircraft were on. He said that he had no idea that it was a passenger aircraft.
The question why nobody on the ground attempted to contact the aircraft still remains. As far as I could find, there is no definitive answer to this. The following is somewhat speculative.
Initially, the USSR believed the aircraft to be a US spy plane (RC-135). While there was some doubt about this (someone even suggested it could be a civilian airliner), the failed attempt to intercept the aircraft over Kamchatka created more confusion among the USSR air defense command personnel. By the time the aircraft approached Sakhalin Island, the situation was already too time critical to further investigate the identity of the aircraft.
That is my interpretation based on the following points of the analysis summary in the report:
3.23 The proximity of an RC-135 (a United States intelligence aircraft) and KE 007 northeast
of Kamchatka Peninsula resulted in confusion and the assumption by the USSR air defence that the
aircraft proceeding towards the USSR was an RC-135.
3.24 USSR military aircraft attempted to intercept KE 007 over Kamchatka Peninsula.
3.25 Information was freely available to flight crews that an aircraft penetrating prohibited
areas of USSR sovereign airspace over Kamchatka Peninsula and Sakhalin Island might be fired upon
without warning.
3.26 The USSR air defence command centre personnel on Sakhalin Island were concerned with
the position of the intruder aircraft in relation to USSR sovereign airspace as well as its identity.
3.27 The time factor became paramount in the USSR air defence command centres. as the
intruder aircraft was about to coast out from Sakhalin Island.
3.28 Exhaustive efforts to identify the intruder aircraft were not made, although apparently
some doubt remained regarding its identity.
