Trump's push for the wall reflected a continuation of the anti-immigrant views that helped fuel his election, some Democrats said.
Trump announced his intention to replace Tillerson with Pompeo, chief of the Central Intelligence Agency (CIA), on March 13.
Trump's remarks came a day after his meeting with Kim Yong Chol, vice chairman of the DPRK's ruling Korean Workers' Party Central Committee, at the White House.
Trump is eager to return to the campaign trail with the election less than four weeks away.
Trump threat to hit back at French tax prompts Macron warning on EU action
Trump has been invited to meet Emperor Naruhito and attend an imperial banquet with the new emperor at the Imperial Palace on Monday.
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Trump has continued to privately discuss the matter of prosecuting his longtime adversaries, including talk of a new special counsel to investigate both Clinton and Comey, the newspaper said, citing two people who had spoken to Trump about the matter.
Trump also spoke of rising violent crime in Chicago and suggested that Democratic presidential candidate Joe Biden wouldn't address it.
Trump on Wednesday said the fate of the landmark summit with Kim will be decided "next week", saying, "We will know next week about Singapore. And if we go, I think it will be a great thing for North Korea."
Tuanyuan, an official online platform named after the Chinese word for reunion, releases information about missing children and was developed for the authorities by e-commerce giant Alibaba.