Update: Goebbels Statements from 1941

Goebbels' diary [hereafter TB], January 31, 1941: "Discussed with Bouhler the discreet liquidation of the mentally ill. 40,000 have gone, 60,000 have still to go. It’s a tough but necessary job. And it has to be done. Bouhler is the right man to do it [Longerich, Peter. Goebbels: A Biography (Kindle Locations 21516-21520). Random House Publishing Group, 2015. Kindle Edition, 21519, n.62]."

Das Reich, July 20, 1941: "Just as the fist of an awakened Germany has struck this racial filth, the fist of an awakened Europe will surely follow. Mimicry will not help the Jews then. They will have to face their accusers. The court of the nations will judge their oppressor.
Without pity or forgiveness, the blow will strike. The world enemy will fall, and Europe will have peace [source and source]."

TB, August 7, 1941: Goebbels states that Jews in the Warsaw ghetto are carriers of infectious diseases and must be locked up and left to themselves or liquidated.

TB August 11, 1941: “In the major cities the Jews are receiving their just deserts. Masses of them are being beaten to death by the self-defense organizations of the Baltic peoples. What the Führer prophesied is coming true: that if Jewry succeeded in once again provoking a war it would cease to exist [Longerich, 21520-21534, n.68]."

TB August 19, 1941, summarized by Longerich, 11185:

As Goebbels had previously assumed, Hitler too adopted an uncompromising position on the “Jewish question.” “He agrees that we should introduce a large visible Jewish badge for all Jews in the Reich” so that “the danger of Jews being grumblers and malcontents without people recognizing them as Jews is removed.” Moreover, Hitler had promised “to deport the Berlin Jews as quickly as possible to the east as soon as the first transportation becomes available. There they will be subjected to a harsher climate.” Later Hitler once more returned to the topic and expressed his conviction that “the prophecy that he had made to the Reichstag back then that, if the Jews succeeded once again in provoking a world war, the final annihilation of the Jews would be assured. During these weeks and months, that was coming true with a certainty that was almost uncanny. The Jews must pay the bill in the east; in Germany they had almost paid it and would have to pay still more in the future.”

TB, August 19, 1941, quoting Hitler: "But I will not rest or be idle until we too have gone all the way with the Jews [translation in Browning, p.320]." Context: The Rumanians had been shooting entire Jewish communities, based on an order given by Antonescu on June 16, 1941 [see Ancel, here, note 21]. Hitler was alluding to a matching escalation in the USSR, with the possibility of also killing Reich Jews after deporting them to the USSR.

TB, August 23, 1941: Goebbels bemoans the fact that the euthanasia program has become too controversial in view of Galen's opposition. Goebbels writes that it is It is intolerable that during a war, hundreds of thousands of people who are completely stupid and can never can be cured are allowed to use up valuable resources.

TB, September 5, 1941: Notes shooting of Jews in Bessarabia [Longerich, 21534, n.68]

TB, October 19, 1941: “massive shooting of Jews in the Ukraine [Longerich, 21534, n.68].”

27-31 October 1941, summarized by Longerich, 11438-11445: 

Goebbels’s solution was to launch another anti-Semitic campaign at the end of October without referring to the deportations from Germany. This campaign once more targeted the alleged dominant influence of the Jews in the Soviet Union, in the United States, and in Great Britain and was intended to prove the existence of the Jewish world conspiracy. Another event, however, formed the prelude: A letter written by the Romanian head of state, Ion Antonescu, to Wilhelm Filderman, the leading Jewish representative in that country, in which he strongly rejected the latter’s complaints about the deportation of the Bessarabian Jews to Transnistria, was given widespread coverage in the press. The press was instructed to give this letter and the deportations from Bessarabia prominence and to recall Hitler’s prophecy of January 1939 in which he had predicted “the annihilation of the Jewish race in Europe” in the event of a world war. The Völkischer Beobachter reported on October 27 under the headline “They Dug Their Own Grave! Jewish Warmongers Sealed Jewry’s Fate.” As instructed by the propaganda minister, the article included the quotation from Hitler’s speech of January 30, 1939, in full and added: “What the Führer announced prophetically then has now become reality. The war of revenge against Germany stirred up by the Jews has now turned on the Jews themselves. The Jews must follow the path that they prepared for themselves.”

Das Reich, November 16, 1941: "The Jews are receiving a penalty that is certainly hard, but more than deserved. World Jewry erred in adding up the forces available to it for this war, and now is gradually experiencing the destructon that it planned for us, and would have carried out without a second thought if it had possessed the ability. It is perishing according its own law: “An eye for an eye, a tooth for a tooth [source and source]."

TB, December 13, 1941, on Hitler's address to the Gau and Reich leaders: "As concerns the Jewish question, the Führer is determined to make a clean sweep. He had prophesied to the Jews that if they once again brought about a world war they would experience their own extermination. This was not just an empty phrase. The World War is there, the extermination of Jewry must be the necessary consequence. This question must be seen without sentimentality. We are not here in order to have sympathy with the Jews, rather we sympathize with our own German people. If the German people have now once again sacrificed as many as 160,000 dead in the Eastern campaign, then the authors of this bloody conflict must pay with their lives [source, paragraph, 5.128]."

Conclusions:

1) Hitler's Reichstag prophecy is invoked repeatedly, and knowingly. Whilst this does not yet entail systematic extermination in death camps, the prospect of mass death being inflicted on deported Jews in the USSR is already a motivating force, albeit for after the war.

2) Antonescu's influence on Hitler has arguably been neglected by historians. Antonescu had initiated the killing of women and children in the USSR via his instructions of June 16, 1941 [see Ancel, here, note 21]. Hitler was aware of these killings via the reports of Einsatzgruppe D. His approval of these killings was an important green light for the Final Solution. Hitler's response to Antonescu's rebuke of Filderman in October 1941, with the Fuehrer again invoking the prophecy, shows that the genocide committed by the Rumanians in Transnistria cohered with Hitler's conception of how his prophecy was now relevant to the times.