Pence had been scheduled to depart on Tuesday night for Cairo. Instead, the trip will be rescheduled for the week of Jan. 14, officials told reporters.
"The vice president is committed to seeing the tax cut through to the finish line," said Alyssa Farah, a spokeswoman for Pence. "The vice president looks forward to traveling to Egypt and Israel in January."
Pence has been a key figure in the Republican effort to overhaul U.S. tax law. He could provide a tie-breaking vote in the Senate if needed, though on Monday it looked like bill has enough votes among Senate Republicans to pass.
Pence was to have spent three days in the region with stops in Cairo and Jerusalem, the first high-level official to visit after Trump reversed decades of U.S. policy and recognized Jerusalem as the capital of Israel.
That decision led to uproar and protest in the region.
The status of Jerusalem, which holds Muslim, Jewish and Christian holy sites, is one of the thorniest obstacles to a peace deal between Israel and the Palestinians, who were furious over Trump's move and have declined to meet with Pence. The international community does not recognize Israeli sovereignty over the full city.
White House officials said the delay was not related to the reaction in the region to Trump's decision.
Two Senate Republican holdouts agreed on Monday to support the tax package. Republicans hold a 52-48 majority in the body.
"The tax vote is still in very good shape, but we don't want to take any chances," a White House official said, adding there were "several difference scenarios" in which Pence might be needed.
The timing of the vote made the trip tricky.
The House of Representatives was due to vote first at around 1:30 p.m. (1830 GMT) on Tuesday, according to Republican aides, while the Senate vote is expected to follow either later on Tuesday or on Wednesday.
White House officials determined that if Pence could not leave for his trip by Tuesday evening, it would have pushed his meeting with Egyptian President Abdel Fattah al-Sisi on Wednesday to a very late hour and left little time to hold meetings with Prime Minister Benjamin Netanyahu and others in Israel before Friday at sunset, when Shabbat, the Jewish Sabbath, begins.
"We ran into time constraints to get it all done before Shabbat," the official said.
Even with the postponement to January, the trip will be overshadowed by Trump's decision on Jerusalem.
Israel considers Jerusalem its eternal capital, while Palestinians want the capital of an independent state of theirs to be in the city's eastern sector, which Israel captured in the 1967 Middle East war and annexed in a move never recognized internationally.
Pence, an evangelical Christian who planned to highlight the plight of Christian minorities during his trip, was not scheduled to meet with Palestinian Christians or with officials from the Coptic Christian church, who declined a meeting in response to the U.S. move. Palestinian President Mahmoud Abbas also refused a meeting with Pence.