This is known to have happened sometimes.
From the Führer:
Hitler no longer appreciated the scale of these exceptional requests. As early as 14 January 1939 he criticised the numerous applications for betterment before the top representatives of the NSDAP. Witness to this "telling off" in the new Reich Chancellery was Günter Kaufmann, then regional leader and head of the Reich Youth Leadership. Hitler said:
"I get wash-basket proof of such applications from you, my party comrades. You obviously know more decent Jews than there are Jews in the German Reich. This is an outrage! I vigorously resist such requests!
— John M. Steiner & Jobst Freiherr von Cornberg: "Willkür in der Willkür. Befreiungen von den antisemitischen Nürnberger Gesetzen", Vierteljahreshefte für Zeitgeschichte, Vol 46, No 2, 1998. (VfZ, own translation, PDF)
Later from Himmler, this time not ear-witness/hearsay, but on tape:
And then along they all come, all the 80 million upright Germans, and each one has his decent Jew. They say: all the others are swine, but here is a first-class Jew.
— The Complete Text of the Poznan Speech, October 4, 1943, Poznan, Poland
It perhaps not superfluous to emphasise very strongly that this is so much hyperbole — as to just calling it a lie is equally appropriate. So, while it did happen in some cases, and some high-ranking nazis complaining about the amount of such exemptions, the reality was of course that the very vast majority of Jews did not find a nazi-party member who was a (former) friend and took action on their behalf. To be clear: there were no noteworthy numbers getting "exception from the final solution". As Himmler made explicit: such occasional request? That's what the SS was there for. To ideally not grant any.
On August 8, 1941, Dr. Werner Feldscher of the Jewish Department of the Reich Ministry of the Interior wrote in a memorandum to Ministerialrat Kaibel:
"The Führer has repeatedly decided that exemptions for racial Jews from the provisions applying to them must be rejected without exception and without regard to the special circumstances of individual cases."
And in a letter of 7 February 1942 from the Reich Chancellery to the Reich Minister of Finance, it was stated:
"The Führer, on other occasions, has expressed his wish that in future no exceptions should be made to the treatment of Jews as provided for by law or by administrative measures, regardless of the reasons for such exceptional treatment.
The letter also contains a reference to a single exception, Mrs Melitta Hoffman, a fully Jewish woman. The applicant had rendered outstanding services to the "movement" before coming to power.
All other applications by so-called full Jews were rejected in the preliminary proceedings and not even submitted to Hitler.
Whether it was 'Goebbels complaining that his office would be receiving too many requests':
He had received personally some of these requests, true. But that tale is rather unlikely for the alleged volume. His office was overseeing quite different matters. The well known sentence "Who is a Jew, I decide" is attributed to Hermann Göring. But in reality he did not have much competence or say in that matter.
The final decision for this arbitrariness officially lay solely on Hitler's desk, where he would judge about such matters mainly via photographs. Indeed in contrast to Göring Goebbels was in a few cases known to issue such orders that some people would get special exemptions. He did so a few times for useful looking cultural actors, independent from Hitler. That a few more 'got away' without much appeal to those three party members was then the result of some second tier officials who delayed a few procedures, while the official stance was still 'when in doubt, assume not Jew' (or those various degrees they created for classification of races). That stance diminished ever further as time went on.
(— Volker Koop: "Wer Jude ist, bestimme ich". "Ehrenarier" im Nationalsozialismus", Böhlau Verlag: Köln, 2014. doi)
The total number of such exemptions, granted until April 1943, with the help of a tiny minority of nazi officials granting really big favours or just sabotaging some procedures of asserting racial status, is guesstimated to be smaller than 1300 individuals altogether. All others suffered the well known fate.